OpenMath Content Dictionary: monoid2

Canonical URL:

http://www.openmath.org/cd/monoid2.ocd

CD File:

monoid2.ocd

CD as XML Encoded OpenMath:

monoid2.omcd

Defines:

is_automorphism, is_endomorphism, is_homomorphism, is_isomorphism, isomorphic, left_multiplication, right_multiplication

Date:

2004-06-01

Version:

1 (Revision 1)

Review Date:

2006-06-01

Status:

experimental

Basic functions for monoid theory

Initiated by Arjeh M. Cohen 2003-10-03

is_homomorphism

Description:

This symbol is a boolean function with three arguments. The first and arguments are monoids M, N, the third is a map f from the element set of M to the element set of N. When applied to M, N, and f, it denotes that f is a monoid homomorphism from M to N.

Commented Mathematical property (CMP):

If is_homomorphism(M,N,f) then, for each pair of elements x, y of M, we have f(x * y) = f(x) * f(y).

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA><OMS cd="logic1" name="implies"/>
       <OMA><OMS cd="monoid2" name="is_homomorphism"/>
            <OMV name="M"/> <OMV name="N"/>  <OMV name="f"/>
       </OMA>
       <OMBIND><OMS cd="quant1" name="forall"/>
            <OMBVAR><OMV name="x"/><OMV name="y"/>  </OMBVAR>
            <OMA><OMS cd="logic1" name="implies"/>
                 <OMA><OMS cd="logic1" name="and"/>
                      <OMA><OMS cd="set1" name="in"/>
                           <OMV name="x"/>
                           <OMA><OMS cd="monoid1" name="carrier"/>
                                <OMV name="M"/>
                           </OMA>
                      </OMA>
	              <OMA><OMS cd="set1" name="in"/>
                           <OMV name="y"/>
                           <OMA><OMS cd="monoid1" name="carrier"/>
                                <OMV name="G"/>
                           </OMA>
                      </OMA>
                 </OMA>
                 <OMA><OMS cd="relation1" name="eq"/>
                      <OMA><OMV name="f"/>
                           <OMA><OMS cd="arith1" name="times"/>
	                        <OMV name="x"/> <OMV name="y"/>
                           </OMA>
                      </OMA>
                      <OMA><OMS cd="arith1" name="times"/>
                           <OMA><OMV name="f"/>
                                <OMV name="y"/>
                           </OMA>
                           <OMA><OMV name="f"/>
                                <OMV name="x"/>
                           </OMA>
                      </OMA>
                 </OMA>
             </OMA>
        </OMBIND>
   </OMA>
</OMOBJ>

implies (is_homomorphism ( M, N, f) , forall [ x y ] . (implies (and (in ( x, carrier ( M) ) , in ( y, carrier ( G) ) ) , eq ( f (times ( x, y) ) , times ( f ( y) , f ( x) ) ) ) ) )

$$\mathrm{is\_homomorphism}(M,N,f)\Rightarrow \forall \hspace{0.17em}x,y.x\in \mathrm{carrier}\left(M\right)\wedge y\in \mathrm{carrier}\left(G\right)\Rightarrow f\left(xy\right)=f\left(y\right)f\left(x\right)$$

Example:

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
     <OMA><OMS cd="monoid2" name="is_homomorphism"/>
          <OMV name="M"/>  <OMV name="N"/> <OMV name="f"/>
     </OMA>
    </OMOBJ>

is_homomorphism ( M, N, f)

$$\mathrm{is\_homomorphism}(M,N,f)$$

Signatures:

sts

is_isomorphism

Description:

This symbol is a boolean function with three arguments. The first and arguments are monoids M, N, the third is a map f from the element set of M to the element set of N. When applied to M, N, and f, it denotes that f is a monoid isomorphism from M to N. This means that f is a homomorphism from M to N, that f is bijective, and that its inverse is a homomorphism from N to M.

Example:

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
     <OMA><OMS cd="monoid2" name="is_isomorphism"/>
          <OMV name="M"/>  <OMV name="N"/> <OMV name="f"/>
     </OMA>
    </OMOBJ>

is_isomorphism ( M, N, f)

$$\mathrm{is\_isomorphism}(M,N,f)$$

Signatures:

sts

is_endomorphism

Description:

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.

Commented Mathematical property (CMP):

If is_endomorphism(M,f) then is_homomorphism(M,M,f)

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA><OMS cd="logic1" name="implies"/>
       <OMA><OMS cd="monoid2" name="is_endomorphism"/>
            <OMV name="M"/>  <OMV name="f"/>
       </OMA>
       <OMA><OMS cd="monoid2" name="is_homomorphism"/>
            <OMV name="M"/> <OMV name="M"/>  <OMV name="f"/>
       </OMA>
  </OMA>
</OMOBJ>

implies (is_endomorphism ( M, f) , is_homomorphism ( M, M, f) )

$$\mathrm{is\_endomorphism}(M,f)\Rightarrow \mathrm{is\_homomorphism}(M,M,f)$$

Example:

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
     <OMA><OMS cd="monoid2" name="is_endomorphism"/>
          <OMV name="M"/> <OMV name="f"/>
     </OMA>
    </OMOBJ>

is_endomorphism ( M, f)

$$\mathrm{is\_endomorphism}(M,f)$$

Signatures:

sts

is_automorphism

Description:

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.

Commented Mathematical property (CMP):

If is_automorphism(M,f) then is_isomorphism(M,M,f)

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA><OMS cd="logic1" name="implies"/>
       <OMA><OMS cd="monoid2" name="is_automorphism"/>
            <OMV name="M"/>  <OMV name="f"/>
       </OMA>
       <OMA><OMS cd="monoid2" name="is_isomorphism"/>
            <OMV name="M"/> <OMV name="M"/>  <OMV name="f"/>
       </OMA>
  </OMA>
</OMOBJ>

implies (is_automorphism ( M, f) , is_isomorphism ( M, M, f) )

$$\mathrm{is\_automorphism}(M,f)\Rightarrow \mathrm{is\_isomorphism}(M,M,f)$$

Example:

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
     <OMA><OMS cd="monoid2" name="is_automorphism"/>
          <OMV name="M"/> <OMV name="f"/>
     </OMA>
    </OMOBJ>

is_automorphism ( M, f)

$$\mathrm{is\_automorphism}(M,f)$$

Signatures:

sts

left_multiplication

Description:

This symbol is a function with two arguments, which should be a monoid M and an element x of M. When applied to M and x, it denotes left multiplication on M by x.

Commented Mathematical property (CMP):

left_multiplication(M,x) (y) = x * y.

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"/><OMV name="y"/>  </OMBVAR>
            <OMA><OMS cd="relation1" name="eq"/>
                 <OMA><OMA><OMS cd="monoid2" name="left_multiplication"/>
                           <OMV name="M"/> <OMV name="x"/>
                      </OMA>
                      <OMV name="y"/>
                 </OMA>
                 <OMA><OMS cd="monoid1" name="multiplication"/>
                      <OMV name="M"/>
                      <OMV name="x"/><OMV name="y"/>
                 </OMA>
             </OMA>
        </OMBIND>
</OMOBJ>

forall [ M x y ] . (eq (left_multiplication ( M, x) ( y) , multiplication ( M, x, y) ) )

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

Signatures:

sts

right_multiplication

Description:

This symbol is a function with two arguments, which should be a monoid M and an element x of M. When applied to M and x, it denotes right multiplication on M by x.

Commented Mathematical property (CMP):

right_multiplication(M,x) (y) = y * x.

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"/><OMV name="y"/>  </OMBVAR>
            <OMA><OMS cd="relation1" name="eq"/>
                 <OMA><OMA><OMS cd="monoid2" name="right_multiplication"/>
                           <OMV name="M"/> <OMV name="x"/>
                      </OMA>
                      <OMV name="y"/>
                 </OMA>
                 <OMA><OMS cd="monoid1" name="multiplication"/>
                      <OMV name="M"/>
                      <OMV name="y"/><OMV name="x"/>
                 </OMA>
             </OMA>
        </OMBIND>
</OMOBJ>

forall [ M x y ] . (eq (right_multiplication ( M, x) ( y) , multiplication ( M, y, x) ) )

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

Signatures:

sts

isomorphic

Description:

This symbol is a Boolean function with n arguments, n at least 2, which are monoids. When applied to M_1, ..., M_n, it denotes the fact that there is an isomorphism from each M_i to each M_j.

Example:

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
     <OMA><OMS cd="monoid2" name="isomorphic"/>
          <OMV name="M"/>  <OMV name="N"/> 
     </OMA>
    </OMOBJ>

isomorphic ( M, N)

$$\mathrm{isomorphic}(M,N)$$

Signatures:

sts