OpenMath Content Dictionary: monoid3

Canonical URL:

http://www.openmath.org/cd/monoid3.ocd

CD File:

monoid3.ocd

CD as XML Encoded OpenMath:

monoid3.omcd

Defines:

automorphism_group, concatenation, cyclic_monoid, direct_power, direct_product, emptyword, free_monoid, left_regular_representation, maps_monoid, strings

Date:

2004-06-01

Version:

3 (Revision 2)

Review Date:

2006-06-01

Status:

experimental

Monoid constructions

Initiated by Arjeh M. Cohen 2003-10-02
Edited AMC 2004-03-05
Edited AMC 2004-06-27

cyclic_monoid

Description:

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.

Example:

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
     <OMA>
      <OMS cd="monoid3" name="cyclic_monoid"/>
      <OMV name="k"/>
      <OMV name="l"/>
     </OMA>
    </OMOBJ>

cyclic_monoid ( k, l)

$$\mathrm{cyclic\_monoid}(k,l)$$

Commented Mathematical property (CMP):

The size of cyclickl(k,l) equals k+l.

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA><OMS cd="relation1" name="eq"/>
       <OMA><OMS cd="set1" name="size"/>
            <OMA><OMS cd="monoid1" name="carrier"/>
                 <OMA><OMS cd="monoid3" name="cyclic_monoid"/>
                      <OMV name="k"/>   <OMV name="l"/>
                 </OMA>
            </OMA>
       </OMA>
       <OMA><OMS cd="arith1" name="plus"/>
            <OMV name="k"/> <OMV name="l"/>
      </OMA>
 </OMA>
</OMOBJ>

eq (size (carrier (cyclic_monoid ( k, l) ) ) , plus ( k, l) )

$$\mathrm{size}\left(\mathrm{carrier}\left(\mathrm{cyclic\_monoid}(k,l)\right)\right)=k+l$$

Signatures:

sts

maps_monoid

Description:

This is a unary function whose argument must be a set X or a positive integer. When applied to X, it refers to the monoid of all functions from X to X if X is a set and to {1,...,X} if X is an integer, whose binary operation is composition of maps and whose identity element is the identity map on the set X, respectively {1,...,X}.

Signatures:

sts

left_regular_representation

Description:

This is a unary function whose argument must be a monoid M. When applied to M, it represents the map from M to the maps monoid on M that assigns to m left multiplication by m on M.

Commented Mathematical property (CMP):

The left regular representation on M applied to the element x of M represents left multiplication by x on M

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
       <OMBIND><OMS cd="quant1" name="forall"/>
            <OMBVAR><OMV name="M"/> <OMV name="x"/>  </OMBVAR>
            <OMA><OMS cd="relation1" name="eq"/>
                 <OMA><OMA><OMS cd="monoid3" name="left_regular_representation"/>
                           <OMV name="M"/>
                      </OMA>
                      <OMV name="x"/>
                 </OMA>
                 <OMA><OMS cd="monoid2" name="left_multiplication"/>
                      <OMV name="M"/> <OMV name="x"/>
                 </OMA>
             </OMA>
        </OMBIND>
</OMOBJ>

forall [ M x ] . (eq (left_regular_representation ( M) ( x) , left_multiplication ( M, x) ) )

$$\forall \hspace{0.17em}M,x.\left(\mathrm{left\_regular\_representation}\left(M\right)\right)\left(x\right)=\mathrm{left\_multiplication}(M,x)$$

Commented Mathematical property (CMP):

The left regular representation is a homomorphism of monoids from M to the maps monoid on M.

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
       <OMBIND><OMS cd="quant1" name="forall"/>
            <OMBVAR><OMV name="M"/>  </OMBVAR>
            <OMA><OMS cd="monoid2" name="is_homomorphism"/>
                 <OMV name="M"/>
                 <OMA><OMS cd="monoid3" name="maps_monoid"/>
                      <OMV name="M"/>
                 </OMA>
                 <OMA><OMS cd="monoid3" name="left_regular_representation"/>
                      <OMV name="M"/>
                 </OMA>
            </OMA>
        </OMBIND>
</OMOBJ>

forall [ M ] . (is_homomorphism ( M, maps_monoid ( M) , left_regular_representation ( M) ) )

$$\forall \hspace{0.17em}M.\mathrm{is\_homomorphism}(M,\mathrm{maps\_monoid}\left(M\right),\mathrm{left\_regular\_representation}\left(M\right))$$

Signatures:

sts

automorphism_group

Description:

This is a function with a single argument which must be a monoid. It refers to the automorphism group of its argument.

Signatures:

sts

direct_product

Description:

This is an n-ary function whose arguments must be monoids. It refers to the direct product of its arguments.

Signatures:

sts

direct_power

Description:

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.

Signatures:

sts

free_monoid

Description:

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.

Example:

The free monoid on the letters a, b:

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA><OMS cd="monoid3" name="free_monoid"/>
       <OMA><OMS cd="list1" name="list"/>
            <OMV name="a"/>  <OMV name="b"/>
       </OMA>
  </OMA>
</OMOBJ>

free_monoid (list ( a, b) )

$$\mathrm{free\_monoid}\left((a,b)\right)$$

Signatures:

sts

strings

Description:

This symbol represents a unary function. The argument is a list or a set. When evaluated on such an argument, the function represents the set of all strings whose characters are entries of the list or set.

Signatures:

sts

emptyword

Description:

This symbol represents a constant. It represents the empty string.

Signatures:

sts

concatenation

Description:

This symbol represents a binary concatenation operation on strings.

Signatures:

sts