added operations for a module as an algebra

PackageInfo.g

@@ -8,8 +8,8 @@ SetPackageInfo( rec(
 
 PackageName := "XModAlg",
 Subtitle := "Crossed Modules and Cat1-Algebras",
-Version := "1.25",
-Date := "17/05/2024", # dd/mm/yyyy format
+Version := "1.25dev",
+Date := "17/06/2024", # dd/mm/yyyy format
 License := "GPL-2.0-or-later",
 
 Persons := [
@@ -81,7 +81,7 @@ PackageDoc := rec(
 ),
 
 Dependencies := rec(
-  GAP := ">=4.11.0",
+  GAP := ">=4.12.0",
   NeededOtherPackages := [ ["XMod", ">=2.87"], 
                            ["LAGUNA", ">=3.9.3"] ],
   SuggestedOtherPackages := [ ],    
@@ -107,7 +107,7 @@ Keywords := ["crossed module of algebras", "cat1-algebra"],
 AutoDoc := rec(
     TitlePage := rec(
         Copyright := Concatenation(
-            "&copyright; 2014-2022, Zekeriya Arvasi and Alper Odabas. <P/>\n", 
+            "&copyright; 2014-2024, Zekeriya Arvasi and Alper Odabas. <P/>\n", 
             "The &XModAlg; package is free software; you can redistribute it ", 
             "and/or modify it under the terms of the GNU General ", 
             "Public License as published by the Free Software Foundation; ", 

doc/algebra.xml

@@ -2,7 +2,7 @@
 <!--                                                                     -->
 <!--  algebra.xml           XModAlg documentation             Z. Arvasi  -->
 <!--                                                        & A. Odabas  -->
-<!--  Copyright (C) 2014-2024, Z. Arvasi & A. Odabas,              	 --> 
+<!--  Copyright (C) 2014-2024, Z. Arvasi & A. Odabas,                    -->
 <!--  Osmangazi University, Eskisehir, Turkey                            --> 
 <!--                                                                     -->
 <!-- ------------------------------------------------------------------- -->
@@ -305,7 +305,15 @@ gap> [ Image(b2,m)=m^3, b2=b1^2 ];
 </Example>
 
 
-<!-- Add other actions here? -->
+<#Include Label="AlgebraModules">
+
+<#Include Label="ModuleAsAlgebra">
+
+<#Include Label="IsModuleAsAlgebra">
+
+<#Include Label="ModuleToAlgebraIsomorphism">
+
+<#Include Label="AlgebraActionByModule">
 
 
 <ManSection>

doc/cat1.xml

@@ -108,10 +108,12 @@ are used with particular choices of algebra homomorphisms.
          Arg="C" />
    <Attr Name="Boundary" Label="for cat1-algebras"
          Arg="C" />
-   <Attr Name="Size2d" Label="for cat1-algebras"
+   <Attr Name="Size2d" Label="for 2d-algebras"
+         Arg="C" />
+   <Attr Name="Dimension" Label="for 2d-algebras"
          Arg="C" />
 <Description> 
-These are the eight main attributes of a pre-cat<M>^{1}</M>-algebra. 
+These are the nine main attributes of a pre-cat<M>^{1}</M>-algebra. 
 <P/> 
 In the example we use homomorphisms between <C>A2c6</C> and <C>I2c6</C> 
 constructed in section <Ref Sect="algebra-homomorphism-lists"/>. 
@@ -131,6 +133,8 @@ gap> IsCat1Algebra( C4 );
 true
 gap> Size2d( C4 );
 [ 64, 8 ]
+gap> Dimension( C4 );
+[ 6, 3 ]
 gap> Display( C4 );
 
 Cat1-algebra [..=>..] :- 

doc/convert.xml

@@ -2,7 +2,7 @@
 <!--                                                                     -->
 <!--  convert.xml          XModAlg documentation              Z. Arvasi  -->
 <!--                                                        & A. Odabas  -->
-<!--  Copyright (C) 2014-2021, Z. Arvasi & A. Odabas,                    --> 
+<!--  Copyright (C) 2014-2024, Z. Arvasi & A. Odabas,                    --> 
 <!--  Osmangazi University, Eskisehir, Turkey                            --> 
 <!--                                                                     -->
 <!-- ------------------------------------------------------------------- -->
@@ -65,18 +65,16 @@ instead of the operation <Ref Oper="Cat1AlgebraSelect"/>.
 These operations are used for constructing a cat<M>^{1}</M>-algebra 
 from a given crossed module of algebras. 
 As an example we use the crossed module <C>XAB</C> constructed in 
-<Ref Oper="XModAlgebraByIdeal"/> 
-(The output from <C>Display</C> needs to be improved.) 
+<Ref Oper="XModAlgebraByIdeal"/>. 
 </Description>
 </ManSection> 
 
 <Example>
 <![CDATA[
 gap> CAB := Cat1AlgebraOfXModAlgebra( XAB );
-[Algebra( GF(5), [ v.1, v.2, v.3, v.4, v.5 ] ) -> A(l,m)]
+[A(l,m) |X A(m)=>A(l,m)]
 gap> Display( CAB );
-
-Cat1-algebra [..=>A(l,m)] :- 
+Cat1-algebra [A(l,m) |X A(m)=>A(l,m)] :- 
 :  range algebra has generators:
 
 [ 
@@ -85,13 +83,40 @@ Cat1-algebra [..=>A(l,m)] :-
   [ [ 0*Z(5), Z(5)^0, Z(5)^3 ], [ 0*Z(5), 0*Z(5), Z(5)^0 ], 
       [ 0*Z(5), 0*Z(5), 0*Z(5) ] ] ]
 : tail homomorphism maps source generators to:
+
+[ 
+  [ [ Z(5)^0, 0*Z(5), 0*Z(5) ], [ 0*Z(5), Z(5)^0, 0*Z(5) ], 
+      [ 0*Z(5), 0*Z(5), Z(5)^0 ] ], 
+  [ [ 0*Z(5), Z(5)^0, Z(5)^3 ], [ 0*Z(5), 0*Z(5), Z(5)^0 ], 
+      [ 0*Z(5), 0*Z(5), 0*Z(5) ] ], 
+  [ [ 0*Z(5), 0*Z(5), Z(5)^0 ], [ 0*Z(5), 0*Z(5), 0*Z(5) ], 
+      [ 0*Z(5), 0*Z(5), 0*Z(5) ] ], 
+  [ [ 0*Z(5), 0*Z(5), 0*Z(5) ], [ 0*Z(5), 0*Z(5), 0*Z(5) ], 
+      [ 0*Z(5), 0*Z(5), 0*Z(5) ] ], 
+  [ [ 0*Z(5), 0*Z(5), 0*Z(5) ], [ 0*Z(5), 0*Z(5), 0*Z(5) ], 
+      [ 0*Z(5), 0*Z(5), 0*Z(5) ] ] ]
+: head homomorphism maps source generators to:
+
+[ 
+  [ [ Z(5)^0, 0*Z(5), 0*Z(5) ], [ 0*Z(5), Z(5)^0, 0*Z(5) ], 
+      [ 0*Z(5), 0*Z(5), Z(5)^0 ] ], 
+  [ [ 0*Z(5), Z(5)^0, Z(5)^3 ], [ 0*Z(5), 0*Z(5), Z(5)^0 ], 
+      [ 0*Z(5), 0*Z(5), 0*Z(5) ] ], 
+  [ [ 0*Z(5), 0*Z(5), Z(5)^0 ], [ 0*Z(5), 0*Z(5), 0*Z(5) ], 
+      [ 0*Z(5), 0*Z(5), 0*Z(5) ] ], 
+  [ [ 0*Z(5), Z(5)^0, Z(5)^3 ], [ 0*Z(5), 0*Z(5), Z(5)^0 ], 
+      [ 0*Z(5), 0*Z(5), 0*Z(5) ] ], 
+  [ [ 0*Z(5), 0*Z(5), Z(5)^0 ], [ 0*Z(5), 0*Z(5), 0*Z(5) ], 
+      [ 0*Z(5), 0*Z(5), 0*Z(5) ] ] ]
 : range embedding maps range generators to:
   [ v.1, v.2 ]
 : kernel has generators:
-  Algebra( GF(5), [ v.4, v.5 ] )
+  [ v.4, v.5 ]
 ]]>
 </Example>
 
+<#Include Label="Cat1AlgebraOfXModAlgebra">
+
 <ManSection>
    <Oper Name="XModAlgebraOfCat1Algebra"
          Arg="C" />

doc/xmod.xml

@@ -2,7 +2,7 @@
 <!--                                                                     -->
 <!--  xmod.xml            XModAlg documentation               Z. Arvasi  -->
 <!--                                                        & A. Odabas  -->
-<!--  Copyright (C) 2014-2022, Z. Arvasi & A. Odabas,                    --> 
+<!--  Copyright (C) 2014-2024, Z. Arvasi & A. Odabas,                    --> 
 <!--  Osmangazi University, Eskisehir, Turkey                            --> 
 <!--                                                                     -->
 <!-- ------------------------------------------------------------------- -->
@@ -215,8 +215,9 @@ gap> SetName( S, "<e5>" );
 gap> RS := Cartesian( R, S );; 
 gap> SetName( RS, "GF(2^2)[k4] x <e5>" ); 
 gap> act := AlgebraAction( R, RS, S );;
-gap> bdy := AlgebraHomomorphismByFunction( S, R, r->r );
-MappingByFunction( <e5>, GF(2^2)[k4], function( r ) ... end )
+gap> bdy := AlgebraHomomorphismByImages( S, R, [e5], [e5] );
+[ (Z(2)^0)*<identity> of ...+(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2 ] -> 
+[ (Z(2)^0)*<identity> of ...+(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2 ]
 gap> IsAlgebraAction( act ); 
 true
 gap> IsAlgebraHomomorphism( bdy );
@@ -238,24 +239,7 @@ Crossed module [<e5>->GF(2^2)[k4]] :-
 ]]>
 </Example>
 
-<ManSection>
-   <Oper Name="XModAlgebraByModule"
-         Arg="M R" />
-<Description> 
-Let <M>M</M> be a <M>R</M>-module.  
-Then <M>\mathcal{X} = (0:M\rightarrow R)</M> is a crossed module. 
-Conversely, given a crossed module 
-<M>\mathcal{X} = (\partial :M\rightarrow R)</M>,
-one can get that <M>\ker\partial</M> is a <M>(R/\partial M)</M>-module.
-<P/>
-</Description>
-</ManSection>
-
-<Example>
-<![CDATA[
-gap> ## example needed 
-]]>
-</Example>
+<#Include Label="XModAlgebraByModule">
 
 <ManSection>
    <Attr Name="Source" Label="for crossed modules of commutative algebras"
@@ -290,9 +274,13 @@ The following standard &GAP; operations have special &XModAlg; implementations:
 <C>Display(X)</C> is used to list the components of <M>\mathcal{X}</M>; 
 </Item> 
 <Item>
-<C>Size2d(X)</C> is used for calculating the order of the
-crossed module <M>\mathcal{X}</M>; 
-</Item> 
+<C>Size2d(X)</C> for a crossed module <M>\mathcal{X}</M> 
+returns a <M>2</M>-element list, the sizes of the source and range, 
+</Item>
+<Item>
+<C>Dimension(X)</C> for a crossed module <M>\mathcal{X}</M> 
+returns a <M>2</M>-element list, the dimensions of the source and range,
+</Item>
 <Item>
 <C>Name(X)</C> is used for giving a name to the crossed module 
 <M>\mathcal{X}</M> by associating the names of source and range algebras. 
@@ -309,7 +297,10 @@ We also show usage of the attributes listed above.
 <Example>
 <![CDATA[
 gap> f := Boundary( XIAk4 );
-MappingByFunction( I(GF5[k4]), GF5[k4], function( i ) ... end )
+[ (Z(5)^2)*<identity> of ...+(Z(5)^0)*f1, (Z(5)^2)*<identity> of ...\
++(Z(5)^
+    0)*f2 ] -> [ (Z(5)^2)*<identity> of ...+(Z(5)^0)*f1, 
+  (Z(5)^2)*<identity> of ...+(Z(5)^0)*f2 ]
 gap> Print( RepresentationsOfObject(XIAk4), "\n" ); 
 [ "IsComponentObjectRep", "IsAttributeStoringRep", "IsPreXModAlgebraObj" ]
 gap> props := [ "CanEasilyCompareElements", "CanEasilySortElements", 

examples/cat1.g

@@ -1,24 +1,30 @@
-#############################################################################
+############################################################################
 ##
-#W  cat1.g                 XMODALG example files               Zekeriya Arvasi
-#W                                                                & Alper Odabas
-#Y  Copyright (C) 2014-2018, Zekeriya Arvasi & Alper Odabas,  
+#W  cat1.g               XMODALG example files               Zekeriya Arvasi
+#W                                                            & Alper Odabas
+#Y  Copyright (C) 2014-2024, Zekeriya Arvasi & Alper Odabas,  
 ##
 
-Print("\nXModAlg test file cat1.g (version 10/11/15) :-");
+Print("\nXModAlg test file cat1.g :-");
 Print("\ntesting constructions of cat1-algebras\n\n");
 
-A := GroupRing(GF(2),Group((1,2,3)(4,5)));
-R := GroupRing(GF(2),Group((1,2,3)));
-f := AllHomsOfAlgebras(A,R);
-g := AllHomsOfAlgebras(R,A);
-C4 := PreCat1Obj(f[6],f[6],g[8]);
-IsCat1Algebra(C4);
-Size(C4);
-Display(C4);
+LoadPackage( "xmodalg" ); 
+
+A := GroupRing( GF(2), Group((1,2,3)(4,5)) );
+SetName( A, "GF(2)[c6]" );
+R := GroupRing( GF(2), Group( (1,2,3) ) );
+SetName( R, "GF(2)[c3]" );
+f := AllAlgebraHomomorphisms( A, R );
+g := AllAlgebraHomomorphisms( R, A );
+C4 := PreCat1AlgebraObj( f[6], f[6], g[8] );
+IsCat1Algebra( C4 );
+Size2d( C4 );
+Dimension( C4 );
+Display( C4 );
 
 C2 := Cat1AlgebraSelect( 4, 6, 2, 2 );
-Size( C2 ); 
+Size2d( C2 );
+Dimension(C2); 
 Display( C2 ); 
 C := Cat1AlgebraSelect(11);
 C := Cat1AlgebraSelect(4,12);
@@ -47,17 +53,19 @@ SetName( R, "GF(2^2)[k4]" );
 e5 := Elements(R)[5]; 
 S := Subalgebra( R, [e5] ); 
 SetName( S, "<e5>" );
-RS := Cartesian( R, S );; 
-SetName( RS, "GF(2^2)[k4] x <e5>" ); 
-act := AlgebraAction( R, RS, S );;
-bdy := AlgebraHomomorphismByFunction( S, R, r->r );
-IsAlgebraAction( act ); 
-IsAlgebraHomomorphism( bdy );
-XM := PreXModAlgebraByBoundaryAndAction( bdy, act );
-CXM := Cat1AlgebraByXModAlgebra( XM );
-Display(CXM);
-X3 := XModAlgebraByCat1Algebra( C3 ); 
-Display( X3 ); 
+
+## the following may be included once AlgebraAction2 has been fixed
+## RS := Cartesian( R, S );; 
+## SetName( RS, "GF(2^2)[k4] x <e5>" ); 
+## act := AlgebraAction( R, RS, S );;
+## bdy := AlgebraHomomorphismByImages( S, R, [e5], [e5] );
+## IsAlgebraAction( act ); 
+## IsAlgebraHomomorphism( bdy );
+## XM := PreXModAlgebraByBoundaryAndAction( bdy, act );
+## CXM := Cat1AlgebraByXModAlgebra( XM );
+## Display(CXM);
+## X3 := XModAlgebraByCat1Algebra( C3 ); 
+## Display( X3 ); 
 
 #############################################################################
 ##

examples/module3.g

@@ -0,0 +1,73 @@
+############################################################################
+##
+#W  module3.g            XModAlg example files                 Chris Wensley 
+## 
+
+LoadPackage( "xmodalg" );
+
+## Chapter 2, Section 2.2.3
+m := [ [0,1,0], [0,0,1], [1,0,0] ];
+Print( "m = ", m, "\n" );
+A3 := Algebra( Rationals, [m] );
+SetName( A3, "A3" );;
+V3 := Rationals^3;;
+M3 := LeftAlgebraModule( A3, \*, V3 );
+SetName( M3, "M3" );
+famM3 := ElementsFamily( FamilyObj( M3 ) );;
+v := [3,4,5];;
+v2 := ObjByExtRep( famM3, v );
+Print( "v = ", v, ",  v2 = ", v2, "\n" );
+Print( "m*v2 = ", m*v2, "\n" );
+genM3 := GeneratorsOfLeftModule( M3 );;
+u2 := 6*genM3[1] + 7*genM3[2] + 8*genM3[3];
+u := ExtRepOfObj( u2 );
+Print( "u2 = ", u2, ",  u = ", u, "\n" );
+
+## Chapter 2, Section 2.2.4
+D3 := LeftActingDomain( M3 );;
+T3 := EmptySCTable( Dimension(M3), Zero(D3), "symmetric" );;
+B3a := AlgebraByStructureConstants( D3, T3 );
+Print( "B3a = ", B3a, "\n" );
+Print( "B3a has generators: ", GeneratorsOfAlgebra( B3a ), "\n" );
+B3 := ModuleAsAlgebra( M3 );             
+Print( "B3 has name: ", Name( B3 ), "\n" );
+Print( "B3 has generators: ", GeneratorsOfAlgebra( B3 ), "\n" );
+
+## Chapter 2, Section 2.2.5
+Print( "IsModuleAsAlgebra( B3 )? ", IsModuleAsAlgebra( B3 ), "\n" );
+Print( "IsModuleAsAlgebra( A3 )? ", IsModuleAsAlgebra( A3 ), "\n" );
+
+## Chapter 2, Section 2.2.6
+Print( "the known attributes of B3 are:\n" );
+Print( KnownAttributesOfObject( B3 ), "\n" );
+M2B3 := ModuleToAlgebraIsomorphism( B3 );
+Print( "M2B3 = ", M2B3, "\n" );
+Print( "Source( M2B3 ) = M3? ", Source( M2B3 ) = M3, "\n" );
+Print( "Source( M2B3 ) = V3? ", Source( M2B3 ) = V3, "\n" );
+B2M3 := AlgebraToModuleIsomorphism( B3 );
+Print( "B2M3 = ", B2M3, "\n" );
+Print( "Range( B2M3 ) = M3? ", Range( B2M3 ) = M3, "\n" );
+Print( "Range( B2M3 ) = V3? ", Range( B2M3 ) = V3, "\n" );
+
+## Chapter 2, Section 2.2.7
+act3 := AlgebraActionByModule( A3, M3 );
+Print( "the action act3 of A3 on B3 is:\n", act3, "\n" );
+a := 2*m + 3*m^2;
+Print( "a = ", a, ":\n" );
+Print( "the image of a under act3 is:\n", Image( act3, a ), "\n" );
+Print( "the image of the action act3 is:\n", Image( act3 ), "\n" );
+Print( "\n" );
+
+## Chapter 4, Section 4.1.7
+X3 := XModAlgebraByModule( A3, M3 );
+Print( "Name( X3 ) = ", Name( X3 ), "\n" );
+Display( X3 );
+
+## Chapter 5, Section 5.1.1
+C3 := Cat1AlgebraOfXModAlgebra( X3 );
+Print( "Name( C3 ) = ", Name( C3 ), "\n" );
+Display( C3 );
+
+############################################################################
+##
+#E  module3.g . . . . . . . . . . . . . . . . . . . . . . . . . . ends here