| Symbol | CD | Description |
| A | setname2 |
This symbol represents the set of algebraic numbers.
|
| above | limit1 |
This symbol is used within a limit construct to show the limit is
being approached from above. It takes no arguments.
|
| abs | arith1 |
A unary operator which represents the absolute value of its
argument. The argument should be numerically valued.
In the complex case this is often referred to as the modulus.
|
| absolute_zero | physical_consts1 |
This symbol represents the absolute zero of temperature, synonymous
with the object of that temperature having zero latent heat.
|
| acceleration | dimensions1 |
This symbol represents the acceleration physical dimension. It is the
second derivative of distance with respect to time.
|
| acre | units_imperial1 |
This symbol represents the measure of one acre. This is a standard
imperial measure for area.
|
| acre_us_survey | units_us1 |
This symbol represents the measure of one U.S. Survey acre.
|
| action | permutation1 | This symbols is
a binary function whose first argument is a permutation (or a endomap)
and whose second argument is a point.
When applied to permutation or endomap p and point x, it represents the image of
the point x under the permutation p.
|
| addition | field1 |
This symbols represents a unary function, whose argument should be a
field. It returns the addition map on the field.
We allow for the map to be n-ary.
|
| addition | ring1 |
This symbols represents a unary function, whose argument should be a
ring. It returns the addition on the ring.
We will allow for the map to be n-ary.
|
| additive_group | field1 |
This symbol is a unary function, whose argument should be a field S.
When applied to S its value is the monoid underlying S.
|
| additive_group | ring1 |
This symbol is a unary function, whose argument should be a ring S.
When applied to S its value is the monoid underlying S.
|
| affine_coordinates | plangeo4 |
This function yields the affine coordinates vector if applied to a point or line with
coordinates in the affine plane.
|
| algorithm | moreerrors |
This symbol represents the error which is returned when an application
raises an error due to algorithmic restrictions of the
implementations. This includes operations not implemented or partially
implemented, divisions by zero and other domain errors. It will have
at least one argument, which is a string describing the problem. It
may have a second argument which is relevant to the error.
|
| alternating_group | group3 | This symbol is a function with one argument, which should be a
set X. When applied to a set X it represents the group of all even
permutations on
X .
|
| alternating_group | permgp2 |
This symbol represents a unary function. Its argument is either a
positive integer or a set.
When evaluated on a set, it represents the
permutation group of all even permutations of that set.
When evaluated on a positive integer n, it represents the
permutation group of all even permutations of the set {1,..., n}.
|
| alternatingn | group3 |
This symbol is a function with one argument, which should be
a natural number n. When applied to n
it represents the group of all even permutations on the set {1,2, ...,n}.
|
| altitude | plangeo3 |
Given a point p and a line L, this defines the segment starting at p
and ending in the unique point of L closest to p.
|
| ambient_ring | polyd1 |
This is a unary function, whose argument should be a DMP f. When
applied to f, it represents the first argument of f, that is,
ring of the form poly_ring_d(...) used to define f.
|
| amp | units_metric1 |
This symbol represents the measure of one amp. This is the standard
SI measure for current.
|
| and | logic1 |
This symbol represents the logical and function which is an n-ary
function taking boolean arguments and returning a boolean value. It
is true if all arguments are true or false otherwise.
|
| angle | plangeo3 |
Angle of a corner, always measured in positive (anti-clockwise) direction.
|
| anonymous | polyd |
Indicates a variable that we do not want to name
|
| anonymous | polyd1 |
Indicates a variable that we do not want to name
|
| anti-Hermitian | linalg5 |
This symbol represents an anti-Hermitian matrix, it takes one
argument. The argument should be a vector of vectors of values which
determine the upper triangle of the matrix. The lower triangle of the
matrix is specified by the following relation: - M^* = transpose(M),
were M^* denotes the matrix consisting of all the complex conjugates
of M. This rules implies that the main diagonal is zero, therefore the
argument should not include it.
|
| antisymmetric | relation0 | Proposition; the type of antisymmetric binary relations.
|
| append | list2 |
The operation of joining one list to another
|
| apply_to_list | fns2 |
This symbol is used to denote the repeated application of an n-ary
function on the elements of a given list. For example when used with
plus or times this can represent sums and products.
The symbol takes two arguments; the first of which is the n-ary
function, the second a list.
|
| approx | relation1 |
This symbol is used to denote the approximate equality of its two arguments.
|
| arc | plangeo3 |
an arc of a circle M from A to B is the set of points of M that are
encountered when traversing the circle clockwise from A to B.
|
| arccos | transc1 |
This symbol represents the arccos function. This is the inverse of the
cos function as described in Abramowitz and Stegun, section 4.4. It
takes one argument.
|
| arccos | transc3 |
This symbol represents the arccos function. This is the multivalued
inverse of the cos function.
|
| arccosh | transc1 |
This symbol represents the arccosh function as described in Abramowitz
and Stegun, section 4.6.
|
| arccosh | transc3 |
This symbol represents the Arccosh function as described in Abramowitz
and Stegun, section 4.6.
|
| arccot | transc1 |
This symbol represents the arccot function as described in Abramowitz
and Stegun, section 4.4.
|
| arccot | transc3 |
This symbol represents the multi-valued arccot function as the inverse of cot
|
| arccoth | transc1 |
This symbol represents the arccoth function as described in Abramowitz
and Stegun, section 4.6.
|
| arccoth | transc3 |
This symbol represents the Arccoth function as described in Abramowitz
and Stegun, section 4.6.
|
| arccsc | transc1 |
This symbol represents the arccsc function as described in Abramowitz
and Stegun, section 4.4.
|
| arccsc | transc3 |
This symbol represents the multivalued arccsc function as the inverse of
csc.
|
| arccsch | transc1 |
This symbol represents the arccsch function as described in Abramowitz
and Stegun, section 4.6.
|
| arccsch | transc3 |
This symbol represents the Arccsch function as described in Abramowitz
and Stegun, section 4.6.
|
| arcsec | transc1 |
This symbol represents the arcsec function as described in Abramowitz
and Stegun, section 4.4.
|
| arcsec | transc3 |
This symbol represents the multivalued arcsec function as the inverse of
sec.
|
| arcsech | transc1 |
This symbol represents the arcsech function as described in Abramowitz
and Stegun, section 4.6.
|
| arcsech | transc3 |
This symbol represents the Arcsech function as described in Abramowitz
and Stegun, section 4.6.
|
| arcsin | transc1 |
This symbol represents the arcsin function. This is the inverse of the
sin function as described in Abramowitz and Stegun, section 4.4. It
takes one argument.
|
| arcsin | transc3 |
This symbol represents the arcsin function. This is the multi-valued inverse
of the sin function as described in Abramowitz and Stegun, section 4.4. It
takes one argument.
|
| arcsinh | transc1 |
This symbol represents the arcsinh function as described in Abramowitz
and Stegun, section 4.6.
|
| arcsinh | transc3 |
This symbol represents the Arcsinh function as described in Abramowitz
and Stegun, section 4.6.
|
| arctan | transc1 |
This symbol represents the arctan function. This is the inverse of the
tan function as described in Abramowitz and Stegun, section 4.4. It
takes one argument.
|
| arctan | transc2 |
This symbol represents the two-argument arctan function as in Fortran's
ATAN2. arctan(x,y) is a value of arctan(y/x). For real x,y arctan(x,y) is
positive when y is positive, negative when y is negative. If y is zero, the
result is 0 if x is positive, and $\pi$ if x is negative. If x is zero, the
result has absolute value $\pi/2$.
|
| arctan | transc3 |
This symbol represents the arctan function. This is the multi-valued
inverse of the tan function.
|
| arctanh | transc1 |
This symbol represents the arctanh function as described in Abramowitz
and Stegun, section 4.6.
|
| arctanh | transc3 |
This symbol represents the Arctanh function as described in Abramowitz
and Stegun, section 4.6.
|
| are_conjugate | group4 |
This symbol represents a boolean ternary function whose first argument is a group G and
whose second and third arguments are elements x and y of G. Its value on G, x,
and y is true if and only if x and y are conjugate in G.
|
| are_distinct | permutation1 |
This symbol is an n-ary boolean function.
When applied to a_1, ..., a_n, it is true if and
only if the arguments are mutually distinct (that
is, a_i and a_j are equal only if i=j).
|
| are_on_circle | plangeo3 |
The statement that a set of points is on one circle.
|
| are_on_conic | plangeo6 |
The symbol is a boolean n-ary function.
Its arguments should be points. When applied to a sequence of points, its
evaluated to true
if and only if there is a conic on which all arguments lie.
|
| are_on_line | plangeo1 |
The statement that a set of points is collinear.
|
| area | dimensions1 |
This symbol represents the area physical dimension.
|
| argument | complex1 |
This symbol represents the unary function which returns the argument
of a complex number, viz. the angle which a straight line drawn from
the number to zero makes with the Real line (measured
anti-clockwise). The argument to the symbol is the complex number whos
argument is being taken.
|
| arrowset | graph1 |
This symbol represents the set of arrows of a directed graph. It takes one argument, the directed graph.
|
| assertion | plangeo1 |
The symbol is a constructor with two arguments.
Its first argument should be a
configuration, its second argument a statement about the
configuration, called thesis.
When applied to a configuration C and a thesis T, the OpenMath object assertion(C,T)
expresses the assertion that T holds in C.
|
| assignment | prog1 |
This symbol is used to assign values to variables. The syntax is
assignment(variable, value), where variable is the encoding of an
OpenMath variable (OMV) and value is an OpenMath object.
|
| associative | semigroup | The type of associative binary operation.
|
| asynchronousError | moreerrors |
This symbol represents the error which is returned when an application
encounters some asynchronous error, for example if a limit in memory
has been reached, or an error has occurred in some system call (I/O
error, disk full, machine down). It should have one argument, which is
a string describing the problem.
|
| atto | units_siprefix1 |
This symbol represents the fact that the subsequent unit has been
effectively multiplied by $10^-18$
|
| attribution | sts |
An `attribution' object consists of pairs of keys and values. The use
of the symbol `attribution' in a signature indicates that the symbol
is to be used as a key.
|
| automorphism_group | group3 |
This is a function with a single argument which must be a group.
It refers to the automorphism group of its argument.
|
| automorphism_group | field4 |
This is a function with a single argument which must be a field.
It refers to the automorphism group of its argument.
|
| automorphism_group | graph2 |
This symbol is a unary function whose argument is an undirected graph.
When applied to an undirected graph G, it represents the automorphism
group of G.
The resulting automorphism group is represented as a permutation group on the
vertices of the graph G.
|
| automorphism_group | magma3 |
This is a function with a single argument which must be a magma.
It refers to the automorphism group of its argument.
|
| automorphism_group | monoid3 |
This is a function with a single argument which must be a monoid.
It refers to the automorphism group of its argument.
|
| automorphism_group | ring5 |
This is a function with a single argument which must be a ring.
It refers to the automorphism group of its argument.
|
| automorphism_group | semigroup3 |
This is a function with a single argument which must be a semigroup.
It refers to the automorphism group of its argument.
|
| automorphism_group | semigroup4 |
This is a function with a single argument which must be a semigroup.
It refers to the automorphism group of its argument.
|
| Avogadros_constant | physical_consts1 |
This symbol represents the number of atoms in 12 grammes of pure
carbon(12). It is approximately 6.0221367*10^(23) +/- 3.6*10^(17).
|
| banded | linalg5 |
This symbol represents a (p,q) banded matrix, it takes one
argument. A (p,q) banded matrix should always be square. The lower non-zero
subdiagonal is the first element of the argument, whilst the highest non-zero
super-diagonal is given by the last element of the argument. The
argument determines the band of possibly non-zero entries which
are positioned around the diagonal. It should be a vector of vectors,
we note that they will not all be the same length, however the length
of the vectors determine p and q. The longest element specifies the
diagonal of the matrix and hence the size of the matrix. Every element
not in the band is zero.
|
| bar | units_imperial1 |
This symbol represents the measure of one bar. This is the standard
imperial measure for pressure.
|
| base | permgp1 |
This is a function with one argument, which should be a permutation group.
When evaluated with argument G it
returns a list of points permuted by G such that the stabilizer of all
elements of the list in G is trivial. Besides, the list is minimal
with respect to the latter property (in the sense that the stabilizer
in G of the elements of no proper
subset is trivial).
|
| based_integer | nums1 |
This symbol represents the constructor function for integers,
specifying the base. It takes two arguments, the first is a positive
integer to denote the base to which the number is represented, the
second argument is a string which contains an optional sign and the
digits of the integer, using 0-9a-z (as a consequence of this no radix
greater than 35 is supported). Base 16 and base 10 are already
covered in the encodings of integers.
|
| Bell | combinat1 |
The Bell numbers: Bell(n) is the total number of possible partitions of a set
of n elements.
|
| below | limit1 |
This symbol is used within a limit construct to show the limit is
being approached from below. It takes no arguments.
|
| big_intersect | set3 |
This symbol is a unary function whose argument should be a collection C of
subsets of a given set. When applied to C, it represents the intersection
over all members of C.
|
| big_union | set3 |
This symbol is a unary function whose argument should be a collection C of
subsets of a given set. When applied to C, it represents the union over all members of C.
|
| bigfloat | bigfloat1 |
The bigfloat constructor takes three arguments, a mantissa, a base and the
exponent.
|
| bigfloatprec | bigfloat1 |
The bigfloat "with precision specified in (another) radix" constructor. Takes
3 arguments, the first argument is a floating point number constructed with the
bigfloat constructor, the second is the new radix, whilst the third specifies
how many digits are significant.
|
| binder | sts |
An `OMBIND' object has three parts: a "binder" such as "lambda" or
"for all", a (list of) bound variables, and an expression. The use of
`binder' in a signature indicates that we are describing something
which can only be used as the first child of an OMBIND construct.
|
| binomial | combinat1 |
The binomial coefficients. binomial(n, m) is the number of ways of choosing m
objects from a collection of n distinct objects without regard to the order.
|
| block | prog1 |
This symbol is meant to represent an arbitray block of code. A block of code
can be empty. The syntax is block(obj1, obj2,...,objN), where obji is the
OpenMath encoding of the ith sentence (or action) inside the body.
|
| Boltzmann_constant | physical_consts1 |
A constant which describes the relationship between temperature and kinetic energy for
molecules in an ideal gas. It is approximately 1.380658*10^(-23)
+/- 1.2*10^(-28) Joules per Kelvin.
|
| Boolean | setname2 |
This symbol represents the set of Booleans. That is the truth values,
true and false.
|
| both_sides | limit1 |
This symbol is used within a limit construct to show the limit is
being approached from both sides. It takes no arguments.
|
| bytearray | omtypes | The type of byte arrays
|
| C | setname1 |
This symbol represents the set of complex numbers.
|
| C | fieldname1 |
This is a symbol representing the field of complex numbers.
|
| calendar_month | units_time1 |
This symbol represents the measure of one month of (calendar) time.
|
| calendar_year | units_time1 |
This symbol represents the measure of one year of (calendar) time.
|
| call_arguments | prog1 |
This symbol can be used to encode the arguments that will be pased to a function
or procedure.
|
| carrier | group1 |
This symbol represents a unary function, whose argument should be a
group G (for instance constructed by group).
When applied to G, its value should be the set of elements of G.
|
| carrier | field1 |
This symbol represents a unary function, whose argument should be a
field S (for instance constructed by field).
When applied to S, its value should be the set of elements of S.
|
| carrier | magma1 |
This symbol represents a unary function, whose argument should be a
magma G (for instance constructed by magma).
When applied to G, its value should be the set of elements of a magma.
|
| carrier | monoid1 |
This symbol represents a unary function, whose argument should be a
monoid M (for instance constructed by monoid).
When applied to M, its value should be the set of elements of a monoid.
|
| carrier | ring1 |
This symbol represents a unary function, whose argument should be a
ring S (for instance constructed by ring).
When applied to S, its value should be the set of elements of S.
|
| carrier | semigroup1 |
This symbol represents a unary function, whose argument should be a
semigroup S (for instance constructed by semigroup). When
applied to S, its value should be the set of elements of S.
|
| cartesian_power | set3 |
This symbol is a binary function whose first argument should be a set A and
whose second argument should be a natural number k.
When applied to A and k, it represents the Cartesian product of k copies of A.
|
| cartesian_product | multiset1 |
This symbol represents an n-ary construction function for constructing
the Cartesian product of multisets. It takes n multiset arguments in order to
construct their Cartesian product.
|
| cartesian_product | set1 |
This symbol represents an n-ary construction function for constructing
the Cartesian product of sets. It takes n set arguments in order to
construct their Cartesian product.
|
| CD | meta |
The top level element for the Content Dictionary. It just acts
as a container for the elements described below.
|
| CDBase | meta |
An optional element.
If it is used it contains a string representing the URI
to be used as the base for generated canonical URI references
for symbols in the CD.
|
| CDComment | meta |
This symbol is used to represent the element of a content dictionary which
explains some aspect of that content dictionary. It should have one string
argument which makes that explanation.
|
| CDComment | metagrp |
This symbol is used to represent the element of a CDGroup which
explains some aspect of the corresponding content dictionary. It
should have one string argument which makes that explanation.
|
| CDDate | meta |
An element which contains a date as a string in the ISO-8601
YYYY-MM-DD format. This gives the date at which the Content Dictionary
was last edited.
|
| CDDefinition | meta |
This symbol is used to represent the element which contains the
definition of each symbol in a content dictionary. That is: it must
contain a 'Name' element and a 'Description' element, and it may contain
an arbitrary number of 'Example', 'FMP' or 'CMP' elements.
|
| CDGroup | metagrp |
This symbol represents the outermost element of a CDGroup. It has an
arbitrary number of arguments which may be elements of type
corresponding to the other symbols defined in this file.
|
| CDGroupDescription | metagrp |
This symbol represents the element of a CDGroup which describes the
CDGroupDescription element. It has one string argument, this should be
the contents of the CDGroupDescription element intended to describe
the mathematical area of the CDGroup.
|
| CDGroupMember | metagrp |
This symbol represents the element of a CDGroup which describes each
CDGroupMember element. It has one string argument, this should be the
contents of the intended CDGroupMember element of the CDGroup. This
should be used to identify each member of the CDGroup.
|
| CDGroupName | metagrp |
This symbol represents the element of a CDGroup which describes the
name of that CDGroup, it has one argument that should be a string
corresponding to the name. The syntactical requirements are given in
the OpenMath standard.
|
| CDGroupURL | metagrp |
This symbol represents the element of a CDGroup which describes the
CDGroupURL element. It has one string argument which should describe
the URL for that CDGroup, not necessarily for the member Content
Dictionaries, The syntactical requirements are given in the OpenMath
standard.
|
| CDGroupVersion | metagrp |
|
| CDName | meta |
An element which contains the string corresponding to the name of the CD.
The string must match the syntax for CD names given in the OpenMath
Standard. Here and elsewhere white space occurring at the beginning or
end of the string will be ignored.
|
| CDName | metagrp |
This symbol represents the element of a CDGroup which describes each
CDName element. It has one string argument, this should be the string
corresponding to the name of a content dictionary which is in this CDGroup.
|
| CDReviewDate | meta |
An element which contains a date as a string in the ISO-8601
YYYY-MM-DD format. This gives the date at which the Content Dictionary
is next scheduled for review. It should be expected to be stable
until at least this date.
|
| CDRevision | meta |
An element which contains a revision number (or minor version number)
This should be a non-negative integer starting from zero for each
new version. Additional examples would be typical changes
to a CD requiring a new revision number.
|
| CDSComment | metasig |
This symbol is used to represent the element of a signature file which
explains some aspect of that signature file. It should have one string
argument which makes that explanation.
|
| CDSignatures | metasig |
This symbol is used to represent the outermost element of the
Signature File which is characterized by two required attributes that
identify the type system and the Content Dictionary whose signatures
are defined. The value of the XML attribute 'type' is the name of the
Content Dictionary or of the CDGroup that represents the type
system. The value of the XML attribute 'cd' is the name of the Content
Dictionary whose symbols are assigned signatures in this Signature
File. It has an arbitrary number of arguments which may be
elements of type corresponding to the other symbols defined in this file.
|
| CDSReviewDate | metasig |
This symbol is used to represent the element of a signature file which
specifies the earliest possible revision date of the signature
file. It should have one string argument which specifies that date. The
date should be in the format YYYY-MM-DD, e.g. 2000-02-29.
|
| CDSStatus | metasig |
This symbol is used to represent the element of a signature file which
specifies the status of that signature file. It should have one
string argument, which should be one of 'official' (approved by the
OpenMath Society according to the procedure outlined in the OpenMath
standard), 'experimental' (currently being tested), 'private' (used by
a private group of OpenMath users) or 'obsolete' (an obsolete
signature file, kept only for archival purposes).
|
| CDStatus | meta |
An element giving information on the status of the CD.
The content of the element must be one of the following strings.
official (approved by the OpenMath Society),
experimental (currently being tested),
private (used by a private group of OpenMath users), or
obsolete (an obsolete CD kept only for archival purposes).
|
| CDURL | meta |
An optional element.
If it is used it contains a string representing the URL where the
canonical reference copy of this CD is stored.
|
| CDURL | metagrp |
This symbol represents the element of a CDGroup which describes each
CDURL element. It has one string argument, this should be the string
corresponding to the contents of the CDURL element for each Content
Dictionary in the CDGroup. The element is optional, in case it is
missing, the location of the CDGroup identified by the element
CDGroupURL is assumed.
|
| CDUses | meta |
An element which contains zero or more CDNames which correspond
to the CDs that this CD depends on, i.e. uses in examples and FMPs. If
the CD is dependent on any other CDs they may be present here.
|
| CDVersion | meta |
An element which contains a version number for the CD.
This should be a non negative integer. Any change to the CD
that affects existing OpenMath applications that support this CD
should result in an increase in the version number.
|
| CDVersion | metagrp |
This symbol represents the element of a CDGroup which describes each
CDVersion element. It has one integral argument, this should specify
which version of the content dictionary is to be taken as member of
the CDGroup. The element is optional. In case it is missing, the last
version is the one included in the CDGroup.
|
| ceiling | rounding1 |
The round up (to +infinity) operation.
|
| center | group3 |
This symbols represents a unary function whose argument should be a group G.
Its value is the biggest subgroup of G all of whose elements
commute with all elements of G.
|
| center | plangeo3 |
Defines the center of a circle.
|
| center_of | plangeo3 |
Gives the center of the circle
|
| center_of_gravity | plangeo3 |
Center of gravity of a number of points.
|
| centi | units_siprefix1 |
This symbol represents the fact that the subsequent unit has been
effectively multiplied by $0.01$
|
| centralizer | group3 |
This symbols represents a binary function whose first argument should be a
group G and whose second argument should be an element g or a list of elements
L of the group G.
Its value is the subgroup of G of all elements
commuting with g or, if the second argument is a list, all elements of L.
|
| characteristic_eqn | linalg4 |
This symbol represents the polynomial which appears in the left hand
side of the characteristic equation of a matrix. It
takes one argument which should be the matrix. A definition of the
characteristic equation is given in Elementary Linear Algebra, Stanley
I. Grossman in Definition 2 of chapter 6, page 535.
|
| charge | dimensions1 |
This symbol represents the charge physical dimension.
|
| circle | plangeo3 |
The symbol represents a circle.
The circle may be subject to constraints.
|
| class | integer2 |
This symbol represents a bivariate function, whose arguments should be integers.
If a, m are integers, then class(a,m) denotes the residue class a mod m in setname2.Zm.
|
| class | polynomial2 |
This symbol represents a bivariate function, whose arguments should be polynomials.
If a, m are polynomials in a polynomial ring R[X], then class(a,m) denotes the residue class a mod m in
the quotient ring R[X]/ (mR[X]).
|
| class | relation3 |
This symbol represents a ternary function whose first argument is a set S,
whose second argument is a relation R on S, and whose third argument is an
element a of S.
When applied to S, R, and a, it represents the set of all elements in S
related to a by R, that is, the set {b in S | (a,b) in R}.
|
| classes | relation3 |
This symbol represents a binary function whose first argument is a set S,
whose second argument is a relation R on S.
When applied to S and R, it represents the set of all elements in S
of the form class(S,R,a) for a in S.
|
| CMP | meta |
An optional element (which may be repeated many times) which contains
a string corresponding to a property of the symbol being
defined.
|
| coefficient | poly |
The coefficient with respect to a list of variables (the second
argument) raised to a list of powers (the third argument).
Zero if no such term is present. Not all variables need be specified.
|
| coefficient | polynomial1 |
This symbol is a binary function whose first argument should be a polynomial
f and whose second argument should be a non-negative integer n.
It represents the coefficient of the i-th power of the variable in the
polynomial f.
|
| coefficient_ring | poly |
The coefficient ring.
|
| coefficient_ring | polynomial1 |
This symbol is a unary function whose argument should be a polynomial.
It represents the coefficient ring of the polynomial.
|
| collect | polyd3 |
This a binary function. Its first argument should be a DMP f, its second
argument a list of positive integers L.
When applied to f and L, it represents the DMP with coefficients from the poly_ring_d
whose variables only have indices i for i not occurring in the list L, and
whose monomials are built up from the variables indexed by the entries
of L.
|
| columncount | linalg4 |
This symbol represents the function which takes one matrix argument
and returns the number of columns in that matrix.
|
| completely_reduced | polyd |
This attribute, attached to a groebnered object, says 'true' if
the base is fully reduced, i.e. no monomial is divisible by the
leading monomial of any other polynomial.
|
| completely_reduced | polygb1 |
This attribute, attached to a groebnered object, says 'true' if
the base is fully reduced, i.e. no monomial is divisible by the
leading monomial of any other polynomial.
|
| complex_cartesian | complex1 |
This symbol represents a constructor function for complex numbers
specified as the Cartesian coordinates of the relevant point on the
complex plane. It takes two arguments, the first is a number x to
denote the real part and the second a number y to denote the imaginary
part of the complex number x + i y. (Where i is the square root of -1.)
|
| complex_cartesian_type | mathmltypes |
A symbol to be used as the argument of the type symbol to convey the
type of a complex number specified in terms of its real and imaginary
parts.
|
| complex_polar | complex1 |
This symbol represents a constructor function for complex numbers
specified as the polar coordinates of the relevant point on the complex
plane. It takes two arguments, the first is a nonnegative number r to
denote the magnitude and the second a number theta (given in radians)
to denote the argument of the complex number r e^(i theta). (i and
e are defined as in this CD).
|
| complex_polar_type | mathmltypes |
A symbol to be used as the argument of the type symbol to convey the
type of a complex number specified in terms of its modulus and argument.
|
| concatenation | monoid3 |
This symbol represents a binary concatenation operation on strings.
|
| concentration | dimensions1 |
This symbol represents the concentration physical dimension, it is the
amount of a substance in a volume.
|
| configuration | plangeo1 |
The symbol represents a configuration in Euclidean
planar geometry consisting of a sequence of geometric objects like points,
lines, etc, but also of other configurations.
|
| conic | plangeo6 |
The symbol represents a conic.
The conic may be subject to constraints.
|
| conjugacy_class | group4 |
This symbol represents a binary function, whose first argument is a group G and
whose second argument is an element x of G. Its value on G and x is the set of elements which
are conjugate to x in G.
|
| conjugacy_class_representatives | group4 |
This symbol represents a unary function whose argument should be a group.
Its value on a group is a set of representatives of the conjugacy classes of
that group.
|
| conjugacy_classes | group4 |
This symbol represents a unary function whose argument should be a group.
Its value on a group is the set of conjugacy classes of
that group.
|
| conjugate | complex1 |
A unary operator representing the complex conjugate of its argument.
|
| conjugation | group2 |
This symbol is a function with two arguments, which should be a group M
and an element x of M.
When applied to M and x, it denotes conjugation on M by x.
|
| conjugation | field2 |
This symbol is a function with two arguments, which should be a field M
and a nonzero element x of M.
When applied to M and x, it denotes conjugation on M by x.
|
| cons | list2 |
This symbol represents the cons list function. It takes 2 arguments:
the second must be a list, where the elements have the same type as
the type of the first. The function denotes a new list which has
the first argument as its first element followed by the elements of
the second argument.
|
| const_node | polyslp |
This constructor takes one argument, which is a value from the
coefficient ring. It is intended to represent a constant node.
|
| constant | linalg5 |
This symbol represents a matrix which has all entries of the same
value. It takes two arguments, the first is the size of the matrix,
the second is the constant which determines every element.
|
| constant_type | mathmltypes |
A symbol to be used as the argument of the type symbol to convey a
type for the common constants, pi ~= 3.1415, e ~= 2.718, i = square
root of -1, gamma ~= .5772, NaN, infinity (all in the nums cd), true
and false (in the logic cd). Also for MathML variables declared to
have type constant, as in <ci type="constant">x</ci>.
|
| convert | poly |
Conversion between polynomial rings. The first argument is a
polynomial and the second is a polynomial ring. This represents the
conversion of the given polynomial as an element of the given ring.
A program that can compute the conversion is required to return
a polynomial in the given ring.
|
| conway_polynomial | finfield1 |
This symbol represents a binary function. Its arguments should be a prime
number p and a positive integer n.
Before defining which of the possible
f(X) is the Conway polynomial we introduce an ordering of the (univariate)
polynomials of degree n over GF(p). Here the coefficients of the polynomials
are taken in {0, ..., p-1}, the indeterminate is X. Let g(X) =
g_nX^n + ... + g_0 and h(X) =
h_nX^n + ... + h_0. Then we define g < h
if and only if there is an index k with g_i = h_i for i
> k and (-1)^{n-k} g_k < (-1)^{n-k}
h_k.
The Conway polynomial f_{p,n}(X) for
GF(p^n) is defined recursively as the smallest polynomial of
degree n with respect to this ordering such that:
1) f_{p,n}(X) is monic,
2) f_{p,n}(X) is primitive, that is, it is irreducible and its
zeros are generators of the
(cyclic) multiplicative group of GF(p^n),
3) for each proper divisor m of n we have that
f_{p,m}(X^{(p^n-1) / (p^m-1)})= 0 mod
f_{p,n}(X); that is, the ((p^n-1) / (p^m-1))-th
power of a zero of f_{p,n}(X) is a zero of f_{p,m}(X).
|
| coordinates | plangeo4 |
This function yields the coordinates vector if applied to a point or line with
coordinates.
|
| coordinatize | plangeo5 |
This symbol is a function of one argument which must be a
configuration or an assertion (as defined in plangeo1).
When applied to a configuration C, it stands for the same
configuration but now with coordinates attached to each object of C.
The new variables are bound within an OMBIND element with head element
the lambda symbol. The bound variables (placed within an OMBVAR
element) are the new variables, and the last argument of OMBIND is
the expression C in which each object is coordinatized.
If an object already has coordinates, these are left as they are. If
not, then new variables are introduced to coordinatize the object.
When applied to an assertion of the form assertion(C,S), it leads to
the same result except that the last argument of OMBIND is the assertion
whose configuration argument is the expression C in which each object
is coordinatized, and whose thesis argument is S.
|
| corner | plangeo2 | The corner between
two halflines L and M, both starting at the same point. Given three
points A, B and C, the corner A, B, C is the corner of the two
halflines BA and BC. Corresponding to the two cases, the symbol can
have as arguments two halflines or three points.
|
| cos | transc1 |
This symbol represents the cos function as described in Abramowitz and
Stegun, section 4.3. It takes one argument.
|
| cosh | transc1 |
This symbol represents the cosh function as described in Abramowitz
and Stegun, section 4.5. It takes one argument.
|
| cot | transc1 |
This symbol represents the cot function as described in Abramowitz and
Stegun, section 4.3. It takes one argument.
|
| coth | transc1 |
This symbol represents the coth function as described in Abramowitz
and Stegun, section 4.5. It takes one argument.
|
| Coulomb | units_metric1 |
This symbol represents the measure of one Coulomb. This is the standard
SI measure for charge.
|
| csc | transc1 |
This symbol represents the csc function as described in Abramowitz and
Stegun, section 4.3. It takes one argument.
|
| csch | transc1 |
This symbol represents the csch function as described in Abramowitz
and Stegun, section 4.5. It takes one argument.
|
| curl | veccalc1 |
This symbol is used to represent the curl function. It takes one
argument which should be a vector of scalar valued functions, intended
to represent a vector valued function and returns a vector of
functions. It should satisfy the defining relation:
curl(F) = i X \partial(F)/\partial(x) + j X \partial(F)/\partial(y) +
j X \partial(F)/\partial(Z) where i,j,k are the unit vectors
corresponding to the x,y,z axes respectively and the multiplication X
is cross multiplication.
|
| current | dimensions1 |
This symbol represents the current physical dimension.
|
| cycle | permutation1 |
This symbol is an n-ary constructor.
It marks a relation on the set of its arguments
a_1, a_2,...,a_n
consisting of the pairs (a_i,a_{i+1}) for i=1,...,n-1
and the pair (a_n,a_1). The arguments a_i should
all be distinct.
The number n is referred to as the length of
the cycle.
|
| cycle_type | permutation1 |
This symbol is a function with one argument,
which is a permutation.
When applied to a permutation P,
it represents the multiset of lengths of cycles
occurring as arguments of P.
|
| cycles | permutation1 |
This symbol has one argument which should be a endomap p. It returns the
list of cycles of p.
|
| cyclic_group | groupname1 |
This symbol is a function with one argument, which should be
a natural number n. When applied to n
it represents the cyclic group of order n.
|
| cyclic_group | permgp2 |
This symbol represents a unary function whose argument should be a positive
integer.
When evaluated at the integer n, it represents the
permutation group generated by the permutation (1,2,...,n).
|
| cyclic_monoid | monoid3 |
This symbol is a function of two natural numbers, the first of which should be
positive. When evaluated at k and l, it
denotes the cyclic monoid with a cycle of length l and a
tail (including the identity element) of length k.
|
| cyclic_semigroup | semigroup3 |
This symbol denotes the cyclic semigroup with a cycle of length l and a
tail of length k.
|
| day | units_time1 |
This symbol represents the measure of one day of time.
The definitions below ignore the possibilities of "leap seconds".
|
| deci | units_siprefix1 |
This symbol represents the fact that the subsequent unit has been
effectively multiplied by $0.1$
|
| decide | directives1 |
This symbol is a function with one argument, which should be a clause.
When applied to a clause, it asks whether
the clause holds.
|
| def_arguments | prog1 |
This symbol can be used to encode the arguments that a function or procedure
can receive.
|
| defint | calculus1 |
This symbol is used to represent definite integration of unary
functions. It takes two arguments; the first being the range (e.g. a
set) of integration, and the second the function.
|
| degree | poly |
The total degree of its argument. The value returned is a
non-negative integer. We note that the degree of 0 is undefined.
Note that this operation takes no account of any weights that have
been defined: see weighted_degree in polyd.
|
| degree | polynomial1 | This symbol represents a
unary function, whose argument should be univariate polynomial. When applied
to a polynomial, it represents its degree, that is the highest power of the
variable occurring in a term of the polynomial. If the polynomial has no
terms, it is the zero polynomial, in which case the value represented is -1.
|
| degree_Celsius | units_metric1 |
This symbol represents the measure of one degree Celsius. This is a standard
metric measure for temperature.
|
| degree_Fahrenheit | units_imperial1 |
This symbol represents the measure of one degree Fahrenheit. This is
the standard imperial measure for temperature.
|
| degree_Kelvin | units_metric1 |
This symbol represents the measure of one degree Kelvin. This is a standard
SI measure for temperature relative to absolute zero.
|
| degree_wrt | poly |
The degree with respect to a variable (the second
argument). We note that the degree of 0 is undefined.
|
| deka | units_siprefix1 |
This symbol represents the fact that the subsequent unit has been
effectively multiplied by $10$
|
| density | dimensions1 |
This symbol represents the density physical dimension, it is the mass
per unit volume.
|
| depth | polyslp |
A unary function taking an slp as argument and returning the
greatest depth of any leaf node, that is the length of the longest
contiguous path to any leaf node.
|
| derived_subgroup | group3 |
The unary function whose value is the subgroup of argument
generated by all products of the form xyx^-1y^-1.
|
| Description | meta |
An element which contains a string corresponding to the
description of either the CD or the symbol
(depending on which is the enclosing element).
|
| determinant | linalg1 |
This symbol denotes the unary function which returns the determinant
of its argument, the argument should be a square matrix.
|
| diagonal_matrix | linalg5 |
This symbol denotes an n_ary function which is used to construct an
(nxn) diagonal matrix, that is a matrix where every non-diagonal
element is zero, the diagonal elements are equal to the n arguments.
|
| diff | calculus1 |
This symbol is used to express ordinary differentiation of a unary
function. The single argument is the unary function.
|
| difference | list3 |
This symbol takes two arguments both a list. It represents a function which returns a list made up of all
the elements of the first list which are not in the second.
|
| digraph | graph1 |
This symbol refers to a digraph. It has two arguments. The first is the set of vertices, the second is the set of arrows. Arrows are represented by lists of length two, where a list represents the arrow from the first element to the second.
|
| dihedral_group | groupname1 | This symbol is a function with one argument, which should be a
positive integer n. When applied to n it represents the dihedral group of
order 2n. This is the group of all isometries (including reflections) of the
regular n-gon in the plane.
|
| dihedral_group | permgp2 |
This symbol represents a unary function whose argument should be a positive
integer.
When evaluated at the integer n, it represents the
dihedral group of all 2n permutations of {1,2,...,n} preserving the n-gon
1,2,...,n.
|
| direct_power | group3 |
This is a binary function whose first argument should be a group
G and whose second argument should be a natural number n.
It refers to the direct product of n copies of G.
|
| direct_power | monoid3 |
This is a binary function whose first argument should be a monoid
M and whose second argument should be a natural number n.
It refers to the direct product of n copies of M.
|
| direct_power | ring3 |
This is a symbol with two arguments.
The first argument should be a ring S
and the second argument a positive integer n.
It denotes the direct product of n copies of S.
|
| direct_power | semigroup3 |
This is a binary function whose first argument should be a semigroup
M and whose second argument should be a natural number n.
It refers to the direct product of n copies of M.
|
| direct_product | group3 |
This is an n-ary function whose arguments must be groups.
It refers to the direct product of its arguments.
|
| direct_product | magma3 |
This is an n-ary function whose arguments must be magmas.
It refers to the direct product of its arguments.
|
| direct_product | monoid3 |
This is an n-ary function whose arguments must be monoids.
It refers to the direct product of its arguments.
|
| direct_product | ring3 |
This is a symbol with two or more arguments, all of which are rings.
It denotes the ring that is the direct product of its arguments.
|
| direct_product | semigroup3 |
This is an n-ary function whose arguments must be semigroups.
It refers to the direct product of its arguments.
|
| discrete_log | finfield1 |
This symbol represents a binary function. The first argument is the
base b, a primitive element of a finite field F. The second argument
is a nonzero element x in F. It returns the smallest nonnegative
integer i such that x=b^i.
|
| discriminant | poly |
Function taking two arguments, it represents the discriminant
of a polynomial, which is the first argument, with
respect to the given variable which is the second argument.
|
| displacement | dimensions1 |
This symbol represents the spatial difference between two points.
The direction of the displacement is taken into account as well as the
distance between the points.
|
| disprove | directives1 |
This symbol is a function with one argument, which should be a clause.
When applied to a clause C, it asks for a
proof of that C does not hold.
|
| distance | plangeo3 |
The distance between two affine points is the Euclidean distance.
The distance between two geometric objects O and O' is the infimum of the
distances between two affine points, one on O and one on O'.
|
| divergence | veccalc1 |
This symbol is used to represent the divergence function. It takes one
argument which should be a vector of scalar valued functions,
intended to represent a vector valued function and returns a
scalar value. It should satisfy the defining relation:
divergence(F) = \partial(F_(x_1))/\partial(x_1) + ...
+ \partial(F_(x_n))/\partial(x_n)
|
| divide | arith1 |
This symbol represents a (binary) division function denoting the first argument
right-divided by the second, i.e. divide(a,b)=a*inverse(b). It is the
inverse of the multiplication function defined by the symbol times in this CD.
|
| divide | opnode |
A constant value, constructs the divide for division nodes.
|
| divides | integer2 |
This symbol represents a bivariate Boolean function, whose arguments should be integers.
When applied to integers a and b, it denotes the property that a divides b.
|
| divides | polynomial2 |
This symbol represents a bivariate Boolean function, whose arguments should be
polynomials in the same polynomial ring.
When applied to a and b, it denotes the property that a divides b.
|
| divisor_of | monoid1 |
This symbol is a ternary function. Its first argument should be a
monoid M and the second and third arguments should be elements of M.
When applied to M, a, and b, it denotes the fact that a is a divisor
of b in M. This means that there are u,v in carrier(M) such that
uav=b.
|
| DMP | polyd |
The constructor of DMPs. The first argument is the polynomial
ring containing the polynomial and the second is a "SDMP".
Should be of the form DMP(PolyRingD(...), SDMP(...))
|
| DMP | polyd1 |
The constructor of DMPs. The first argument is the polynomial
ring containing the polynomial and the second is a "SDMP".
Should be of the form DMP(poly_ring_d(...), SDMP(...))
|
| DMPL | polyd |
The constructor for lists of multivariate polynomial members of the
same polynomial ring. The first argument is a polynomial ring
and the rest are "SDMP"s. DMPL can be attributed with the "ordering"
symbol to indicate a particular ordering for monomials of all its
polynomials.
Should be of the form DMPL(PolyRingD(...), SDMP(...)+)
|
| DMPL | polyd1 |
The constructor for lists of multivariate polynomial members of the
same polynomial ring. The first argument is a polynomial ring
and the rest are "SDMP"s. DMPL can be attributed with the "ordering"
symbol to indicate a particular ordering for monomials of all its
polynomials.
Should be of the form DMPL(poly_ring_d(...), SDMP(...)+)
|
| domain | fns1 |
This symbol denotes the domain of a given function, which is the set of
values it is defined over.
|
| domain | permutation1 |
This symbol is a function with one argument which is a endomap. When
applied to a endomap whose arguments are a_1,...,a_n, it represents the
set {1,...,n}.
|
| domainofapplication | fns1 |
The domainofapplication element denotes the domain over which a given
function is being applied. It is intended in MathML to be a more general
alternative to specification of this domain using
such quantifier elements as bvar, lowlimit or condition.
|
| e | nums1 |
This symbol represents the base of the natural logarithm, approximately 2.718.
See Abramowitz and Stegun, Handbook of Mathematical Functions,
section 4.1.
|
| edgeset | graph1 |
This symbol represents the set of edges of an undirected graph. It takes one argument, the undirected graph.
|
| eigenvalue | linalg4 |
This symbol represents the eigenvalue of a matrix. It takes two
arguments the first should be the matrix, the second should be an
index to specify the eigenvalue. The ordering imposed on the
eigenvalues is first on the modulus of the value, and second on the
argument of the value. A definition of eigenvalue is
given in Elementary Linear Algebra, Stanley I. Grossman in Definition 1
of chapter 6, page 533.
|
| eigenvector | linalg4 |
This symbol represents the eigenvector of a matrix. It takes two
arguments the first should be the matrix, the second should be an
index to specify which eigenvalue this eigenvector should be paired
with. The ordering is as given in the eigenvalue symbol. A definition
of eigenvector is given in Elementary Linear Algebra, Stanley
I. Grossman in Definition 1 of chapter 6, page 533.
|
| elimination | polyd |
This is an ordering, which is partially in terms of one
ordering, and partially in terms of another.
First argument is a number of variables.
Second is ordering to apply on the first so many variables.
Third is an ordering on the rest, to be used to break ties.
|
| elimination | polyd2 |
This is an ordering, which is partially in terms of one
ordering, and partially in terms of another.
First argument is a number of variables.
Second is ordering to apply on the first so many variables.
Third is an ordering on the rest, to be used to break ties.
|
| emptyset | multiset1 |
This symbol is used to represent the empty multiset, that is the multiset which
contains no members. It takes no parameters.
|
| emptyset | set1 |
This symbol is used to represent the empty set, that is the set which
contains no members. It takes no parameters.
|
| emptyword | monoid3 |
This symbol represents a constant.
It represents the empty string.
|
| encodingError | moreerrors |
This symbol represents the error which is returned when an application
detects a lexical or syntactic error. It should have one argument
which is a string, which should explain the error that occurred.
|
| endomap | permutation1 |
This symbol is an n-ary constructor. Its arguments should
be positive integers. When applied to arguments a_1,...,a_n,
the resulting value is the map sending i to a_i for i=1,...,n.
|
| endomap_left_compose | permutation1 |
This symbol is a binary function. Its arguments should be endomaps with
identical domain D. When applied to arguments P1 and P2, the resulting
value is the endomap which maps x in D to P1(P2(x)).
|
| endomap_right_compose | permutation1 |
This symbol is a binary function. Its arguments should be endomaps with
identical domain D. When applied to arguments P1 and P2, the resulting
value is the endomap which maps x in D to P2(P1(x)).
|
| endpoint | plangeo2 |
The endpoint of a halfline.
|
| endpoints | plangeo2 |
The two endpoints of a segment.
|
| energy | dimensions1 |
This symbol represents the energy physical dimension.
|
| entry | list3 |
This symbol represents a binary function whose first argument should be a list
L and whose second argument should be a positive integer i such that
the absolute value of i is in the interval [1..n], where n is the length of L.
If i is positive, it returns the i-th entry L[i] of L, if i is negative it
returns
the (n+1-i)-th entry of L.
|
| eq | relation1 |
This symbol represents the binary equality function.
|
| eqmod | integer2 |
This symbol represents a Boolean valued trivariate function, whose arguments should be integers.
When applied to integers a, b, m, it denotes the Boolean
evalue of the assertion that a and b are equal modulo m.
|
| eqmod | polynomial2 |
This symbol represents a Boolean valued trivariate function, whose arguments should be polynomials.
When applied to polynomials a, b, m, it denotes the Boolean
evalue of the assertion that a and b are equal modulo m.
|
| eqs | relation4 |
This symbol is used to denote the n-ary version of equality.
When applied to n arguments a1, ..., an, it represents the boolean expression
that
a1, a2, ,,, and an are equal.
|
| equivalence | relation0 | Proposition; the type of equivalence relations,
namely relations that are reflexive, symmetric and transitive.
|
| equivalence_closure | relation3 |
This symbol represents a binary function whose first argument is a set S,
whose second argument is a relation R on S.
When applied to S and R, it represents the smallest equivalence relation
(with respect to inclusion) on S containing R.
|
| equivalent | logic1 |
This symbol is used to show that two boolean expressions are logically
equivalent, that is have the same boolean value for any inputs.
|
| error | sts |
The error symbol is the 'return type' of error symbols in the error
signature file.
|
| euler | integer2 |
This symbol denotes the univariate Euler totient function.
If m is an integer, then euler(m) denotes the order of the multiplicative
group of invertible elements in
the residue class ring Z/mZ.
|
| evaluate | poly |
Evaluation of a polynomial at a value or vector of values.
|
| evaluate | directives1 |
This symbol is a function with one argument, which should be a
mathematical expression.
When applied to a mathematical expression, it asks for an evaluation
or simplification of the expression. The evaluation or simplification
to be carried out by a service is a simpler mathematical expression (in some definition of complexity
of objects) which is equal to the argument.
|
| evaluate_to_type | directives1 |
This symbol is a function with two arguments, which should be
a mathematical expression and a type.
When applied to a mathematical expression E and a type T, it asks for an evaluation
or simplification of E to a mathematical expression of type T.
|
| exa | units_siprefix1 |
This symbol represents the fact that the subsequent unit has been
effectively multiplied by $10^18$
|
| Example | meta |
An element which contains an arbitrary number of children,
each of which is either a string or an OpenMath Object.
These children give examples in natural language, or in OpenMath, of the
enclosing symbol definition.
|
| exists | quant1 |
This symbol represents the existential ("there exists") quantifier
which takes two arguments. It must be placed within an OMBIND element. The first
argument is the bound variables (placed within an OMBVAR element), and the second
is an expression.
|
| exp | transc1 |
This symbol represents the exponentiation function as described in
Abramowitz and Stegun, section 4.2. It takes one argument.
|
| expand | poly |
Converts a factored or squarefreed form into the expanded
polynomial over the same ring, so that factored(recursive)
-> recursive, etc.
|
| expand | polynomial1 |
Expands a polynomial.
|
| expand | polyoperators1 |
Expands a polynomial. Acts as expand(expresion).
|
| explore | directives1 |
This symbol is a unary function whose argument should be
a mathematical assertion.
When applied to an assertion A, it asks for conditions under which the
assertion holds.
|
| expression | group1 |
This symbol is a function with two arguments. Its first
argument should be a group. The
second should be an arithmetic expression A,
whose operators are
times and power, and whose leaves are members of the carrier of G.
When applied to
G and A, it denotes the element (of G) that is obtained from the
leaves of A by applying the multiplication and the power map of G instead of the
times and power
from the CD arith1 appearing in A.
The symbol alg1.one occurring in A will be interpreted as
the identity of G.
|
| expression | field1 |
This symbol is a function with two arguments. Its first
argument should be a field. The
second should be an arithmetic expression A,
whose operators are
times, plus, minus, unary_minus, and power, and whose leaves are members of the carrier of G.
When applied to
G and A, it denotes the element (of G) that is the element obtained from the
leaves of A by applying the operations of G instead of those
from the CD arith1 according to A. Here multiplication, addition, subtraction,
minus, and power take over the roles of
times, plus, minus, unary_minus, and power, respectively.
Also, an integer m occurring in A will be interpreted as a member of G by interpreting it as
the sum of m copies of the identity element, the symbol alg1.one will be
interpreted as the identity,
and the symbol alg1.zero will be
interpreted as the zero of G.
|
| expression | monoid1 |
This symbol is a function with two arguments. Its first
argument should be a monoid. The
second should be an arithmetic expression A,
whose operators are
times and power, and whose leaves are members of the carrier of G.
The second argument of power should be nonnegative. When applied to
G and A, it denotes the element (of G) that is obtained from the
leaves of A by applying the multiplication and the power map of G instead of the
times and power
from the CD arith1 appearing in A.
The symbol alg1.one occurring in A will be interpreted as
the identity of G.
|
| expression | ring1 |
This symbol is a function with two arguments. Its first
argument should be a ring. The
second should be an arithmetic expression A,
whose operators are
times, plus, minus, unary_minus, and power, and whose leaves are members of
the carrier of G.
(Here an integer m will be interpreted as a member of G by interpreting it as
the sum of m copies of the identity element, the symbol alg1.one will be
interpreted as the identity,
and the symbol alg1.zero will be
interpreted as the zero of G.)
When applied to
G and A, it denotes the element (of G) that is the element obtained from the
leaves by applying the arithmetic operations of G instead of those
from the CD arith1.
|
| expression | semigroup1 |
This symbol is a function with two arguments. Its first
argument should be a semigroup G. The
second should be an arithmetic expression A,
whose operators are
times and power, and whose leaves are members of the carrier of G.
The second argument of power should be positive. When applied to
G and A, it denotes the element (of G) that is obtained from the
leaves of A by applying the multiplication and the power map of G instead of the
times and power of the CD arith1 appearing in A.
|
| extended_gcd | arith3 | The symbol represents the n-ary function,
a_1,...,a_n to return a list consisting of
the gcd (greatest common divisor) of its arguments, together with
n elements x_1,...,x_n such that
gcd(a_1,...,a_n)=x_1 a_1+...+x_n a_n
|
| extended_in | polygb2 |
This symbol is a function of at least 3 arguments. The first argument is a list of variables.
The second and third argument are lists of polynomials in the variables from the first
argument, C and T respectively.
When applied to its arguments, it represents the boolean value of the assertion that all elements t
in T can be written as t = f_1*c_1 + ... + f_n*c_n (c_i in C).
If the optional 4th argument is 1, those f_i are returned.
|
| factor | poly |
The decomposition of its argument into irreducible
factors. A program that can compute the factorization is required
to return a "factored" object - see above.
It is currently an open question whether powers of 1 can be omitted.
|
| factor | polyoperators1 |
The action of factoring a polynomial into irreducible factors
(I know this is field dependent but lets keep it simple by now).
|
| factor_of | semigroup1 |
This symbol is a ternary function. Its first argument should be a
semigroup S and the second and third arguments should be elements of
S. When applied to S, a, and b, it denotes the fact that a is a
divisor of b in S. This means that there are u,v in carrier(S) such
that uav=b.
|
| factored | poly |
The constructor for a factorization. Its arguments are formal
powers (see previous operator), where the polynomials are supposed
to be irreducible (except possibly for a content from the ground
ring).
Note that "factored" is not a call to factorise something, rather
a statement that we know a factorisation.
|
| factorial | integer1 |
The symbol to represent a unary factorial function on non-negative integers.
|
| factorof | integer1 |
This is the binary OpenMath operator that is used to indicate the
mathematical relationship a "is a factor of" b, where a is the
first argument and b is the second. This relationship is
true if and only if b mod a = 0.
|
| factors | polynomial3 |
This symbol is a unary function, whose argument should be a polynomial f.
When applied to f, it represents a complete list of irreducible factors of f.
|
| factors | polyoperators1 |
The action of returning a list composed of the irreducible factors of a
polynomial. (I know this is field dependent but lets keep it simple by
now).
|
| false | logic1 |
This symbol represents the boolean value false.
|
| Faradays_constant | physical_consts1 |
This symbol represents the electric charge carried by one mole of
electrons. It is approximately 96485.309 +/- 0.029 Coulombs per mole.
|
| femto | units_siprefix1 |
This symbol represents the fact that the subsequent unit has been
effectively multiplied by $10^-15$
|
| Fibonacci | combinat1 |
The Fibonacci numbers, defined by the linear recurrence:
Fibonacci(0) = 0, Fibonacci(1) = 1, and
Fibonacci(n + 1) = Fibonacci(n) + Fibonacci(n - 1).
Note that some authors define Fibonacci(0) = 1.
|
| field | field1 |
This symbol is a constructor for fields. It takes seven arguments
R, a, o, n, m, e, i: which are, respectively,
a set R to specify the elements in the field,
a binary operation a on R, an element o of R, and a unary
operation n on R such that [R,a,o,n] is a commutative group, a
binary operation m on R, an element e of R, and a map from R - {o}
to itself such that
[R,m,e] is a monoid and such that [R - {o},m',e,i]
is a group, where m' is the restriction of m to R -{o}.
|
| field_by_conway | finfield1 |
This symbol represents a binary function. The first argument should be a
prime number p, the second argument a positive integer n. This symbol returns
the field GF(q)[X]/ (C(X)), where q = p^n, X is an indeterminate, C(X) is the
Conway polynomial f_{n,p}(X), and (C(X)) is the ideal in the
polynomial ring GF(q)[X] generated by C(X).
|
| field_by_poly | field3 |
This symbol is a binary function whose first argument is a univariate
polynomial ring R over a field, and whose second argument is an irreducible
polynomial f in this polynomial ring R. So, when applied to R and f, the
function has value the quotient ring R/(f).
|
| field_by_poly_map | field4 |
Same as quotient_by_poly_map in CD ring5, except that R and the quotient ring R[X]/(f) are
fields (so f is irreducible in R[X]).
|
| field_by_poly_vector | field4 |
This symbol is a binary function. Its first argument should be
a field_by_poly(R,f). Its second argument should be a
list L of elements of F, the coefficient field of the univariate polynomial
ring R = F[X].
The length of the list L should be equal to the degree d of f.
When applied to R and
L, it represents the element L[0] + L[1] x + L[2] x^2 + ... + L[d-1] ^(d-1) of
R/(f),
where x stands for the image of x under the natural quotient map R -> R/(f).
If the first argument is a field_by_conway(p,n), defined in the CD finfield1, then
the same interpretation holds, where R and f are respectively poly_ring_d(GFp(p),1) and conway_polynomial(p,n).
|
| find | directives1 |
This symbol is a binder, whose body should be a clause.
When bound to a variable (or list of variables) x with body P(x), it asks for a
mathematical expression A such that P(A) holds.
|
| first | list2 |
This symbol represents a function which returns the first elements of
its argument, which should be a list.
|
| fix | permutation1 |
This symbol is a function with two arguments. The
first argument should be a permutation, the second
argument a set.
When applied to a permutation g and a set X, it represents
the subset A of X all points that do not belong to the support of g.
|
| float | omtypes | The type of floating point numbers
|
| floor | rounding1 |
The round down (to -infinity) operation.
|
| FMP | meta |
An optional element which contains an OpenMath Object.
This corresponds to a property of the symbol being defined.
|
| fn_type | mathmltypes |
A symbol to be used as the argument of the type symbol to convey the
type for a function name.
|
| foot | units_imperial1 |
This symbol represents the measure of one foot. This is the standard
imperial measure for distance.
|
| foot_us_survey | units_us1 |
This symbol represents the measure of one U.S. Survey foot.
|
| for | prog1 |
This symbol can be used to encode the for loop. The syntax is
for(block1,conditional_block,block3,block4), where block1 is the
inicialization block, conditional_block is the conditional block that
determines the end of the loop, block3 is the incremental block and block4
is the body of the for loop. Each of this blocks should be present (althougth
they can be empty).
|
| forall | quant1 |
This symbol represents the universal ("for all") quantifier which takes two
arguments. It must be placed within an OMBIND element. The first argument is the
bound variables (placed within an OMBVAR element), and the second is an expression.
|
| force | dimensions1 |
This symbol represents the force physical dimension.
|
| fraction_field | field3 |
This is a unary function. Its argument should be a domain (as in CD ring4).
It denotes the fraction field of the domain.
|
| free_field | field3 |
This symbol represents a binary function. The first argument should be a
natural number p which is zero or a prime number,
the second argument a list or a
set L. When evaluated on such arguments p and L, the function represents the
field of rational functions in L over the rationals if p = 0 and over the
field of integers mod p if p is a prime.
|
| free_group | group3 |
This symbol represents a unary function. The argument is a list or a
set. When evaluated on such an argument, the function represents the
free group generated by the entries of the list or set.
|
| free_magma | magma3 |
This symbol represents a binary function. The argument is a
list or a set.
When evaluated on such an argument, the function represents the
free magma generated by the entries of the list or set.
|
| free_monoid | monoid3 |
This symbol represents a unary function. The argument is a list or a
set. When evaluated on such an argument, the function represents the
free monoid generated by the entries of the list or set.
|
| free_ring | ring3 |
This symbol represents a binary function. The first argument should be a ring
and the second a list or a
set. When evaluated on such arguments R and L, the function represents the
free ring over R generated by the elements (or entries) of L.
This ring can also be viewed as the ring of non-commutative polynomials over R
with variables the elements of L.
|
| free_semigroup | semigroup3 |
This symbol represents a binary function. The argument is a list or a
set. When evaluated on such an argument, the function represents the
free semigroup generated by the entries of the list or set.
|
| function | fns3 |
This symbol denotes a function constructor.
When aplied to at least two arguments, which are sets,
the first argument is the domain and the second the range of the function.
When applied to at least three arguments, the first two of which are
stes and the third of which is a lambda expression,
the third argument gives the function specification.
|
| function_block | prog1 |
The block of code defining the body of the function. The syntax is
function_block(local_var,block1), where local_var encodes the local
variables (private to the function body) and block1 is the body of
the function. Both locar_var and block1 should be present (and of
course both can be also empty).
|
| function_call | prog1 |
Symbol function_call can be used to "call" already defined functions.
The syntax is function_call(name, call_arguments), where name is the
encoding of an OpenMath variable (OMV) representing the name of the
function and call_arguments are the arguments to pass to the function.
Both, name and call_arguments, should be present but call_arguments can be
empty.
|
| function_definition | prog1 |
The symbol function_definition can be is used to define a function. The syntax is
function_definition(name, def_arguments, function_block), where name is the
encoding of an OpenMath variable (OMV) representing the name of the funtion,
def_arguments is the enconding of the arguments that the function receives and
function_block is the body of the function (local variables declarations +
body of the function). Functions are completely unaware of the rest of the
"world" except for the information they received from the arguments. Functions
are only allowed to return values by means of the return construct.
|
| gamma | nums1 |
A symbol to convey the notion of the gamma constant
as defined in Abramowitz and Stegun, Handbook of Mathematical
Functions, section 6.1.3. It is the limit of
1 + 1/2 + 1/3 + ... + 1/m - ln m
as m tends to infinity, this is approximately 0.5772 15664.
|
| gas_constant | physical_consts1 |
This symbol represents the constant which is equal to the ratio of the
pressure times the volume and the temperature of an ideal gas. It is
approximately 8.31451 +/- 7.0*10^(-05) Joules per mole per Kelvin.
|
| gcd | arith1 |
The symbol to represent the n-ary function to return the gcd (greatest
common divisor) of its arguments.
|
| gcd | poly |
The n-ary greatest common divisor of its polynomial arguments.
This is unique up to units.
|
| gcd | polynomial3 |
The n-ary greatest common divisor for univariate polynomials over fields.
|
| gcd | polyoperators1 |
The n-ary greatest common divisor for univariate polynomials.
|
| generalized_quaternion_group | groupname1 | This symbol is a function with one argument, which should be a
positive integer. When applied to n it represents the generalized quaternion group
of order 4n. This is the group with three generators a, b, and c and
relations c = a^2 = b^n, c*a = a*c , b*c = c*b, a*b = b*a*c, and c^2 = 1.
|
| generators | permgp1 | This is a function with one argument, which should be a
permutation group. When evaluated with argument G it returns the list
of permutations which occur in the definition of G.
|
| geq | relation1 |
This symbol represents the binary greater than or equal to function
which returns true if the first argument is greater than or equal to
the second, it returns false otherwise.
|
| GFp | setname2 |
This symbol represents the finite field of integers modulo p, where p is a
prime.
|
| GFpn | setname2 |
This symbol represents the finite field with p^n elements, where p is a prime.
|
| giga | units_siprefix1 |
This symbol represents the fact that the subsequent unit has been
effectively multiplied by $10^9$
|
| GL | group3 |
This symbol is a function with one argument, which should be a
vector space or a module V. When applied to
V it represents the group of all invertible linear transformations of V.
|
| GLn | group3 |
This symbol is a function with two arguments. The first should be a positive
integer n, the second a
field F. When applied to
n and F it represents the group of all invertible linear transformations of
the vector space over F of dimension n.
|
| global_var | prog1 |
This symbol can be used to declare global variables as know to function.
|
| grad | veccalc1 |
This symbol is used to represent the grad function. It takes one
argument which should be a scalar valued function and returns a
vector of functions. It should satisfy the defining relation:
grad(F) = (\partial(F)/\partial(x_1), ... ,\partial(F)/partial(x_n))
|
| graded_lexicographic | polyd |
Total degree order, graded with the lexicographic ordering.
Note that, if a poly_ring_d_named is used, lexigographic refers
to the order of the variables in the poly_ring_d_named, not to
their order as strings.
|
| graded_lexicographic | polyd2 |
Total degree order, graded with the lexicographic ordering.
|
| graded_reverse_lexicographic | polyd |
Total degree order, graded with the reverse lexicographic ordering.
Note that, if a poly_ring_d_named is used, lexigographic refers
to the order of the variables in the poly_ring_d_named, not to
their order as strings.
|
| graded_reverse_lexicographic | polyd2 |
Total degree order, graded with the reverse lexicographic ordering.
|
| gramme | units_metric1 |
This symbol represents the measure of one gramme. This is not quite the
standard SI measure for mass, which is the kilogramme, but OpenMath
chooses to regard the gramme as standard, otherwise one would have to call
it the milli-kilogramme.
|
| graph | graph1 |
This symbol represents an undirected graph. It takes two
arguments: the vertex set of the graph and the edge set.
The vertices can be arbitrary OpenMath objects. Each edge should be a set consisting of two vertices.
|
| gravitational_constant | physical_consts1 |
This symbol represents the constant of proportionality in Newtons law
of universal gravitation which states; Two bodies attract each other
with equal and opposite forces; the magnitude of this force is
proportional to the product of the two masses and is also proportional
to the inverse square of the distance between the centers of mass of
the two bodies. It is approximately equal to: 6.672*10^(-11) Newton
square metres per kilogramme squared.
|
| groebner | polyd |
The groebner basis (lt-reduced, minimal) of a set of polynomials,
with respect to a given ordering. First argument is an ordering, the
second is a list of polynomials. A program that can compute
the basis is required to return a "groebnered" object.
|
| groebner | polygb1 |
The groebner basis (reduced, minimal) of a set of polynomials, with
respect to a given ordering. First argument is a list of
variables, the second is an ordering, the
third is a list of polynomials. A program that can compute
the basis is required to return a "groebner_basis" object.
|
| groebner_basis | polygb1 | The
constructor for a Groebner basis (reduced, minimal). The first is a
list of variables, the second argument is an ordering, the third is
the Groebner Basis itself (with respect to the ordering) that should
be represented as a polynomial expression. |
| groebnered | polyd |
The constructor for a Groebner basis (reduced, minimal). The first
argument is an ordering, the second is the Groebner Basis itself
(with respect to the ordering) that should be represented as a DMPL.
|
| groebnered | polygb1 |
The constructor for a Groebner basis (reduced, minimal). The first
argument is an ordering, the second is the Groebner Basis itself
(with respect to the ordering) that should be represented as a DMPL.
|
| group | group1 |
This symbol is a constructor for groups. It takes four arguments in
the following order: a set to specify the elements in the group, a
binary operation to specify the group operation, an element to specify the
identity, and a unary operation to
specify inverses of group elements. Both the binary and unary operations should act on elements
of the set and return an element of the set.
|
| group | permgp1 |
This symbol represents an n-ary function. The first argument is a
group operation
(usually, left_compose or right_compose),
the other n-1 arguments represent permutations.
When evaluated on such arguments, the function represents the
permutation group generated by the last n-1 arguments.
|
| gt | relation1 |
This symbol represents the binary greater than function which returns
true if the first argument is greater than the second, it returns false
otherwise.
|
| H | setname2 |
This symbol represents the set of quaternions.
|
| halfline | plangeo2 |
The halfline starting at A and going through B.
The symbol takes as arguments the points A and B.
|
| hecto | units_siprefix1 |
This symbol represents the fact that the subsequent unit has been
effectively multiplied by $100$
|
| Hermitian | linalg5 |
This symbol represents a Hermitian matrix, it takes one
argument. The argument should be a vector of vectors of values which
determine the upper triangle of the matrix. The lower triangle of the
matrix is specified by the following relation: M^* = transpose(M),
were M^* denotes the matrix consisting of all the complex conjugates
of M.
|
| homomorphism_by_generators | group5 |
This is a function with three arguments the first two of which must be groups
F and K.
The third argument should be a set or a list L of ordered pairs (lists of length 2). Each
pair [x,y] from L consists of an element x from F and an element y from K.
When applied to F, K, and L, the symbol represents the group homomorphism from F
to K that maps the first entry x of each pair [x,y] to the second entry y of the same pair.
|
| homomorphism_by_generators | field4 |
This is a function with three arguments the first two of which must be fields
F and K.
The third argument should be a set or a list L of ordered pairs (lists of length 2). Each
pair [x,y] from L consists of an element x from F and an element y from K.
when applied to F, K, and L, the symbol represents the homomorphism from F
to K that maps the first entry x of each pair [x,y] to the second entry y of the same pair.
|
| homomorphism_by_generators | ring5 |
This is a function with three arguments the first two of which must be monoids
F and K.
The third argument should be a set or a list L of ordered pairs (lists of length 2). Each
pair [x,y] from L consists of an element x from F and an element y from K.
when applied to F, K, and L, the symbol represents the monoid homomorphism from F
to K that maps the first entry x of each pair [x,y] to the second entry y of the same pair.
|
| homomorphism_by_generators | semigroup4 |
This is a function with three arguments the first two of which must be semigroups
F and K.
The third argument should be a set or a list L of ordered pairs (lists of length 2). Each
pair [x,y] from L consists of an element x from F and an element y from K.
when applied to F, K, and L, the symbol represents the homomorphism from F
to K that maps the first entry x of each pair [x,y] to the second entry y of the same pair.
|
| hour | units_time1 |
This symbol represents the measure of one hour of time.
|
| i | nums1 |
This symbol represents the square root of -1.
|
| ideal | plangeo5 | This symbol is a function in one argument, which should
be a coordinatized configuration (that is, each
geometric object involved has coordinates).
When evaluated at
a configuration C it represents a function (given by a
lambda binder) mapping the new parameters (arising
when the inequality properties in the configuration
are being translated into polynomials) to a list of
polynomials in the coordinates that are variables
which, when equated to zero, represent conditions
equivalent to those describing the configuration C.
When evaluated at an assertion assertion(C,S) it represents a function (given by a
lambda binder) mapping the new parameters (arising
when the inequality properties in the configuration
are being translated into polynomials) to a list of
polynomials in the coordinates that are variables
which, when equated to zero, represent conditions
equivalent to those describing the configuration C.
|
| ideal | ring3 | This symbol represents a
binary function. The first argument is a ring R and the second argument is a
list or a set. When evaluated on R and such a second argument, the function
represents the ideal in R generated by the entries of the list or set.
|
| identity | fns1 |
The identity function, it takes one argument and returns the same value.
|
| identity | linalg5 |
This symbol denotes a unary function which is used to construct an
(nxn) identity matrix where n is the single positive integral argument.
|
| identity | group1 |
This symbols represents a unary function, whose argument should be a
group. It returns the identity element of the group.
|
| identity | field1 |
This symbols represents a unary function, whose argument should be a
field. It returns the identity element of the field.
|
| identity | monoid1 |
This symbols represents a unary function, whose argument should be a
monoid. It returns the identity element of the monoid.
|
| identity | ring1 |
This symbols represents a unary function, whose argument should be a
ring. It returns the identity element of the ring.
|
| if | prog1 |
The symbol can be used to encode the if, then, else construct. The syntax is
if(conditional_block,block1,block2), where the conditional_block is the block
that determines wich of the block of codes block1 and block2 is going to be
executed, block1 is the then block and block2 if the else block. The
conditional_block and block1 are required but block2 is optional.
|
| image | fns1 |
This symbol denotes the image of a given function, which is the set of
values the domain of the given function maps to.
|
| imaginary | complex1 |
This represents the imaginary part of a complex number
|
| implies | logic1 |
This symbol represents the logical implies function which takes two
boolean expressions as arguments. It evaluates to false if the first
argument is true and the second argument is false, otherwise it
evaluates to true.
|
| in | multiset1 |
This symbol has two arguments, an element and a multiset. It is
used to denote that the element is in the given multiset.
|
| in | set1 |
This symbol has two arguments, an element and a set. It is used to
denote that the element is in the given set.
|
| in | list2 |
This symbol has two arguments, an element and a list. It is used to
denote that the element is in the given list.
|
| in | polygb2 |
This symbol is a function of at least 4 arguments. The first argument
is a polynomial p,
the second is a list of variables, the third is an ordering
the fourth is a list of polynomials B, and,
optionally, the fifth is a polynomial_ring.
When applied to its arguments, it represents the boolean value
of the assertion that p belongs to the ideal generated by B.
|
| in_radical | polygb2 |
This symbol is a function of at least 4 arguments. The first argument
should be a polynomial p,
the second is a list of variables, the third is an ordering
the fourth is a list of polynomials B, and
optionally: the fifth is a polynomial_ring.
When applied to its arguments, it represents the boolean value
of the assertion that p belongs to the radical ideal generated by B.
|
| incident | plangeo1 |
The symbol represents the logical incidence function which is a
binary function taking arguments representing
geometric objects like points and lines and returning a boolean value.
It is true if and only if the first argument is incident to the second.
|
| indNat | indnat | Attribution tag to denote the
type of inductively defined natural numbers. It is also denoted as
setname1:N.
|
| IndType | icc | Constructor for Inductive Types.
Takes arguments the constructor functions for the
inhabitants of the type and their signatures.
|
| infinity | nums1 |
A symbol to represent the notion of infinity.
|
| inp_node | polyslp |
This constructor takes one argument, which is a variable. The return
value is intended to represent an input node.
|
| int | calculus1 |
This symbol is used to represent indefinite integration of unary functions.
The argument is the unary function.
|
| int2flt | coercions | The function that converts an integer to a float.
|
| integer | omtypes | The type of integers
|
| integer_interval | interval1 |
A symbol to denote a discrete 1 dimensional interval from the first
argument to the second (inclusive), where the discretisation occurs at unit
intervals. The arguments are the start and the end points of the interval
in that order.
|
| integer_type | mathmltypes |
A symbol to be used as the argument of the type symbol to convey the
type of an integer.
|
| integers | ring3 |
This is a symbol representing the ring of integers.
|
| intersect | multiset1 |
This symbol is used to denote the n-ary intersection of
multisets. It takes multisets as arguments, and denotes the
multiset that contains all the elements that occur in all of
them, with multiplicity the minimum of their multiplicities
in all multisets.
|
| intersect | set1 |
This symbol is used to denote the n-ary intersection of sets. It takes
sets as arguments, and denotes the set that contains all the
elements that occur in all of them.
|
| interval | interval1 |
A symbol to denote a continuous 1-dimensional interval without any
information about the character of the end points (used in definite
integration). The arguments are the start and the end points of the interval
in that order.
|
| interval_cc | interval1 |
A symbol to denote a continuous 1-dimensional interval with both end
points included in the interval. The arguments are the start and the
end points of the interval in that order.
|
| interval_co | interval1 |
A symbol to denote a continuous 1-dimensional interval with the first
point included in the interval, but the last excluded. The arguments
are the start and the end points of the interval in that order.
|
| interval_oc | interval1 |
A symbol to denote a continuous 1-dimensional interval with the first
point excluded from the interval, but the last included. The arguments
are the start and the end points of the interval in that order.
|
| interval_oo | interval1 |
A symbol to denote a continuous 1-dimensional interval with both end
points excluded from the interval. The arguments are the start and the end
points of the interval in that order.
|
| inverse | fns1 |
This symbol is used to describe the inverse of its argument (a
function). This inverse may only be partially defined because the
function may not have been surjective. If the function is not
surjective the inverse function is
ill-defined without further stipulations. No assumptions are made on
the semantics of this inverse.
|
| inverse | arith2 |
A unary operator which represents the inverse of an element of a set. This
symbol could be used to represent additive or multiplicative inverses.
|
| inverse | field1 |
This symbol represents a unary function, whose argument should be a field S.
It returns the map sending a nonzero element of S to its multiplicative
inverse.
|
| inverse | permutation1 |
This symbol is a unary function. Its
argument should be a permutation. When applied to
argument P, the resulting
value is the inverse permutation of P.
|
| inversion | group1 |
This symbol represents a unary function, whose argument should be a
group G. It returns the map sending an element of G to its inverse.
|
| invertibles | group3 |
This symbol is a function with one argument, which should be
a monoid M. When applied to M
it represents the group of all invertible elements of M.
|
| invertibles | monoid1 |
This symbol is a unary function. Its argument should be a monoid M.
When applied to M, it denotes the submonoid of M consisting of all
invertible elements in M. This is a group.
|
| invertibles | ring3 |
This is a unary function, whose argument is
a ring R. When applied to R,
it denotes the set of invertible elements of R with respect to the
multiplication on R.
|
| irreflexive | relation0 | Proposition; the type of irreflexive binary relations.
|
| is_affine | plangeo4 |
Boolean function testing whether a point or line is affine.
|
| is_associative | magma1 |
The unary boolean function whose value is true iff the argument is an
associative magma.
|
| is_automorphism | group2 |
This symbol is a boolean function with two arguments.
The first is a group M,
the second is a map f from the element set of M to the element set of M.
When applied to M and f, it denotes a group automorphism f of M.
|
| is_automorphism | field2 |
This symbol is a boolean function with two arguments.
The first is a field M,
the second is a map f from the element set of M to the element set of M.
When applied to M and f, it denotes a field automorphism f of M.
|
| is_automorphism | graph2 |
This symbol is a boolean function with two arguments.
The first is a graph M,
the second is a map f from the element set of M to the element set of M.
When applied to M and f, it denotes a graph automorphism f of M.
|
| is_automorphism | magma2 |
This symbol is a boolean function with two arguments.
The first is a magma M,
the second is a map f from the element set of M to the element set of M.
When applied to M and f, it denotes a magma automorphism f of M.
|
| is_automorphism | monoid2 |
This symbol is a boolean function with two arguments.
The first is a monoid M,
the second is a map f from the element set of M to the element set of M.
When applied to M and f, it denotes a monoid automorphism f of M.
|
| is_automorphism | ring2 |
This symbol is a boolean function with two arguments.
The first is a ring M,
the second is a map f from the element set of M to the element set of M.
When applied to M and f, it denotes a ring automorphism f of M.
|
| is_automorphism | semigroup2 |
This symbol is a boolean function with two arguments.
The first is a semigroup M,
the second is a map f from the element set of M to the element set of M.
When applied to M and f, it denotes a semigroup automorphism f of M.
|
| is_bijective | permutation1 |
This symbol has one argument which should be a endomap p. It returns true
if {a_1,...,a_n}={1,...,n} where a_1,...a_n are the arguments of p and
false otherwise.
|
| is_commutative | group1 |
The unary boolean function whose value is true iff the argument is a
commutative group.
|
| is_commutative | field1 |
The unary boolean function whose value is true iff the argument is a
commutative field.
|
| is_commutative | magma1 |
The unary boolean function whose value is true iff the argument is a
commutative magma.
|
| is_commutative | monoid1 |
The unary boolean function whose value is true iff the argument is a
commutative monoid.
|
| is_commutative | ring1 |
The unary boolean function whose value is true iff the argument is a
commutative ring.
|
| is_commutative | semigroup1 |
The unary boolean function whose value is true iff the argument is a
commutative semigroup.
|
| is_coordinatized | plangeo5 |
This symbol is a boolean valued function of one argument which must be a
configuration.
When applied to an argument C, it represent the value true if C is a
configuration such that each object occurring in C (as well as in its
subconfigurations) has coordinates (that is, the set_affine_coordinates field
is present as an argument to the object), and value false otherwise.
If an object already has coordinates, these are left as they are. If
not, then new variables are introduced to coordinatize the object.
|
| is_domain | ring4 |
This symbol represents a boolean
unary function. The argument is a ring R.
When evaluated on R, the function returns true if R is a domain
and false otherwise. A domain is a commutative ring without zero divisors.
|
| is_endomap | permutation1 |
This symbol is an n-ary function. Its arguments should be
positive integers. When applied to arguments a_1,...,a_n,
the resulting value is true if a_i is at most n for all i,
otherwise it is false.
|
| is_endomorphism | group2 |
This symbol is a boolean function with two arguments.
The first argument is a group M,
the second is a map f from the element set of M to the element set of M.
When applied to M and f, it denotes that f is a group endomorphism from M
to M.
|
| is_endomorphism | field2 |
This symbol is a boolean function with two arguments.
The first argument is a field M,
the second is a map f from the element set of M to the element set of M.
When applied to M and f, it denotes that f is a field endomorphism from M
to M.
|
| is_endomorphism | graph2 |
This symbol is a boolean function with two arguments.
The first argument is a graph M,
the second is a map f from the element set of M to the element set of M.
When applied to M and f, it denotes that f is a graph endomorphism from M
to M.
|
| is_endomorphism | magma2 |
This symbol is a boolean function with two arguments.
The first argument is a magma M,
the second is a map f from the element set of M to the element set of M.
When applied to M and f, it denotes that f is a magma endomorphism from M
to M.
|
| is_endomorphism | monoid2 |
This symbol is a boolean function with two arguments.
The first argument is a monoid M,
the second is a map f from the element set of M to the element set of M.
When applied to M and f, it denotes that f is a monoid endomorphism from M
to M.
|
| is_endomorphism | ring2 |
This symbol is a boolean function with two arguments.
The first argument is a ring M,
the second is a map f from the element set of M to the element set of M.
When applied to M and f, it denotes that f is a ring endomorphism from M
to M.
|
| is_endomorphism | semigroup2 |
This symbol is a boolean function with two arguments.
The first argument is a semigroup M,
the second is a map f from the element set of M to the element set of M.
When applied to M and f, it denotes that f is a semigroup endomorphism from M
to M.
|
| is_equivalence | relation3 |
This symbol represents the boolean binary function which returns
true if and only if the second argument is a symmetric relation on the first.
|
| is_field | ring4 |
This is unary boolean function whose argument should be a ring R.
The value is true if and only if the ring is commutative |