added second example of Cat1AlgebraOfXModAlgebra to Ch.5

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" />

init.g

@@ -1,8 +1,8 @@
-#############################################################################
+############################################################################
 ##
 #W  init.g                 The XMODALG package               Zekeriya Arvasi
-#W                                                             & Alper Odabas
-#Y  Copyright (C) 2014-2022, Zekeriya Arvasi & Alper Odabas,  
+#W                                                            & Alper Odabas
+#Y  Copyright (C) 2014-2024, Zekeriya Arvasi & Alper Odabas,  
 ##
 
 if not IsBound( PreImagesRepresentativeNC ) then 
@@ -14,4 +14,3 @@ ReadPackage( "xmodalg", "lib/algebra.gd" );
 ReadPackage( "xmodalg", "lib/module.gd" );
 ReadPackage( "xmodalg", "lib/alg2obj.gd" );  
 ReadPackage( "xmodalg", "lib/alg2map.gd" );  
-

lib/alg2obj.gi

@@ -505,10 +505,6 @@ InstallGlobalFunction( XModAlgebra, function( arg )
     if ( ( nargs = 2 ) and IsAlgebraHomomorphism( arg[1] )
                        and IsAlgebraAction( arg[2] ) ) then
         return XModAlgebraByBoundaryAndAction( arg[1], arg[2] );
-    # module
-    elif ( ( nargs = 2 ) and IsAlgebra( arg[1] )
-                         and IsLeftModule( arg[2] ) ) then 
-        return XModAlgebraByModule( arg[1], arg[2] );
     # ideal
     elif ( ( nargs = 2 ) and IsAlgebra( arg[1] )
             and IsAlgebra( arg[2] ) and IsIdeal(arg[1],arg[2]) ) then 
@@ -523,9 +519,14 @@ InstallGlobalFunction( XModAlgebra, function( arg )
     # Multiplier algebra
     elif ( ( nargs = 1 ) and IsAlgebra( arg[1] )) then
         return XModAlgebraByMultiplierAlgebra( arg[1] );
+    # module
+    elif ( ( nargs = 2 ) and IsAlgebra( arg[1] )
+                         and IsLeftModule( arg[2] ) ) then 
+        return XModAlgebraByModule( arg[1], arg[2] );
+    else
+        # alternatives not allowed
+        Error( "usage: XModAlgebra( bdy, act ); " );
     fi;
-    # alternatives not allowed
-    Error( "usage: XModAlgebra( bdy, act ); " );
 end );
 
 ############################################################################
@@ -1005,29 +1006,40 @@ InstallOtherMethod( Display, "display a pre-cat1-algebra",
 function( C1A )
 
     local  Csrc, Crng, Cker, gensrc, genrng, genker, name,
-           t, h, e, b, k, imt, imh, ime, imb, imk;
+           t, h, e, b, k, imt, imh, ime, imb, imk, eqth;
 
     name := Name( C1A );
     Csrc := Source( C1A );
+    gensrc := GeneratorsOfAlgebra( Csrc );
     Crng := Range( C1A );
-    Cker := Kernel( C1A );
     genrng := GeneratorsOfAlgebra( Crng );
+    Cker := Kernel( C1A );
+    genker := GeneratorsOfAlgebra( Cker );    
     t := TailMap( C1A );
     h := HeadMap( C1A );
-    e := RangeEmbedding( C1A );    
+    e := RangeEmbedding( C1A );
+    imt := List( gensrc, x -> Image( t, x ) );
+    imh := List( gensrc, x -> Image( h, x ) );
+    eqth := imt = imh;
+    ime := List( genrng, x -> Image( e, x ) );
     if HasXModAlgebraOfCat1Algebra(C1A) then
-        gensrc := Csrc;
-        genker := Cker;    
         if IsCat1Algebra( C1A ) then
-            Print( "\nCat1-algebra ", name, " :- \n" );
+            Print( "Cat1-algebra ", name, " :- \n" );
         else
-            Print( "\nPre-cat1-algebra ", name, " :- \n" );
+            Print( "Pre-cat1-algebra ", name, " :- \n" );
         fi;
-        ime := List( genrng, x -> Image( e, x ) );
-
         Print( ":  range algebra has generators:\n" );
         Print( "  ", genrng, "\n" );
-        Print( ": tail homomorphism maps source generators to:\n" );
+        if eqth then 
+            Print( ": tail homomorphism = head homomorphism\n" );
+            Print( "  they map the source generators to:\n" );
+            Print( "  ", imt, "\n" );
+        else
+            Print( ": tail homomorphism maps source generators to:\n" );
+            Print( "  ", imt, "\n" );
+            Print( ": head homomorphism maps source generators to:\n" );
+            Print( "  ", imh, "\n" );
+        fi;
         Print( ": range embedding maps range generators to:\n" );
         Print( "  ", ime, "\n" );
         if ( Size( Cker ) = 1 ) then
@@ -1038,19 +1050,14 @@ function( C1A )
         fi;    
     else
         b := Boundary( C1A );
-        gensrc := GeneratorsOfAlgebra( Csrc );
-        genker := GeneratorsOfAlgebra( Cker );
         k := KernelEmbedding( C1A );
-        imt := List( gensrc, x -> Image( t, x ) );
-        imh := List( gensrc, x -> Image( h, x ) );
-        ime := List( genrng, x -> Image( e, x ) );
         imb := List( genker, x -> Image( b, x ) );
         imk := List( genker, x -> Image( k, x ) );
 
         if IsCat1Algebra( C1A ) then
-            Print( "\nCat1-algebra ", name, " :- \n" );
+            Print( "Cat1-algebra ", name, " :- \n" );
         else
-            Print( "\nPre-cat1-algebra ", name, " :- \n" );
+            Print( "Pre-cat1-algebra ", name, " :- \n" );
         fi;
         Print( ": source algebra has generators:\n" );
         Print( "  ", gensrc, "\n" );
@@ -1077,7 +1084,6 @@ function( C1A )
     #     Print( ": associated crossed module is ", 
     #            XModAlgebraOfCat1Algebra( C1A ), "\n" );
     # fi;
-    Print( "\n" );
 end );
 
 ############################################################################
@@ -1757,7 +1763,7 @@ function( XM )
     h := AlgebraHomomorphismByImages( P, R, vecP, imgs ); 
     t := Projection( P, 1 ); 
     e := Embedding( P, 1 );
-    C := PreCat1AlgebraObj( t, h, e ); 
+    C := PreCat1AlgebraObj( t, h, e );
     return C; 
 end ); 
 

lib/algebra.gi

@@ -780,7 +780,10 @@ InstallMethod( SemidirectProductOfAlgebras,
             SetEntrySCTable( T, i+n1, j+n1, L ); 
         od; 
     od;
-    P := AlgebraByStructureConstants( F, T ); 
+    P := AlgebraByStructureConstants( F, T );
+    if HasName( A1 ) and HasName( A2 ) then
+        SetName( P, Concatenation( Name( A1 ), " |X ", Name( A2 ) ) ); 
+    fi;
     SetSemidirectProductOfAlgebrasInfo( P, rec( algebras := [ A1, A2 ], 
                                                 action := act, 
                                                 embeddings := [ ], 

lib/module.gd

@@ -38,11 +38,11 @@
 ## [ 3, 4, 5 ]
 ## gap> m*v2;
 ## [ 4, 5, 3 ]
-gap> genM3 := GeneratorsOfLeftModule( M3 );;
-gap> u2 := 6*genM3[1] + 7*genM3[2] + 8*genM3[3];
-[ 6, 7, 8 ]
-gap> u := ExtRepOfObj( u2 );
-[ 6, 7, 8 ]
+## gap> genM3 := GeneratorsOfLeftModule( M3 );;
+## gap> u2 := 6*genM3[1] + 7*genM3[2] + 8*genM3[3];
+## [ 6, 7, 8 ]
+## gap> u := ExtRepOfObj( u2 );
+## [ 6, 7, 8 ]
 ## ]]>
 ## </Example>
 ## </Subsection>
@@ -230,8 +230,8 @@ DeclareOperation( "AlgebraActionByModule", [ IsAlgebra, IsLeftModule ] );
 ## </ManSection>
 ## <Example>
 ## <![CDATA[
-## gap> X0 := XModAlgebraByModule( A, M );
-## [ A(M) -> Q[3,3] ]
+## gap> X3 := XModAlgebraByModule( A3, M3 );
+## [ A(M3) -> A3 ]
 ## gap> Display( X3 );
 ## Crossed module [A(M3)->A3] :- 
 ## : Source algebra A(M3) has generators:
@@ -249,3 +249,38 @@ DeclareOperation( "AlgebraActionByModule", [ IsAlgebra, IsLeftModule ] );
 ## <#/GAPDoc>
 ##
 DeclareOperation( "XModAlgebraByModule", [ IsAlgebra, IsLeftModule ] );
+
+############################################################################
+##
+## addition to Cat1AlgebraOfXModAlgebra
+##
+## <#GAPDoc Label="Cat1AlgebraOfXModAlgebra"> 
+## <P/>
+## As a second example, we convert the crossed module <M>X3</M>
+## constructed in <Ref Sect="XModAlgebraByModule"/>
+
+
+## <Example>
+## <![CDATA[
+## gap> C3 := Cat1AlgebraOfXModAlgebra( X3 );
+## [A3 |X A(M3)=>A3]
+## gap> Display( C3 );           
+## Cat1-algebra [A3 |X A(M3)=>A3] :- 
+## :  range algebra has generators:
+##   [ [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] ]
+## : tail homomorphism = head homomorphism
+##   they map the source generators to:
+##   [ [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ], 
+##   [ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ], 
+##   [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ], 
+##   [ [ 0, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ], 
+##   [ [ 0, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ], 
+##   [ [ 0, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ] ]
+## : range embedding maps range generators to:
+##   [ v.1 ]
+## : kernel has generators:
+##   [ v.4, v.5, v.6 ]
+## ]]>
+## </Example>
+## <#/GAPDoc>
+##

read.g

@@ -1,10 +1,8 @@
-#############################################################################
+############################################################################
 ##
 #W  read.g                 The XMODALG package               Zekeriya Arvasi
-#W                                                             & Alper Odabas
-##  version 1.19, 16/05/2021 
-##
-#Y  Copyright (C) 2014-2021, Zekeriya Arvasi & Alper Odabas,  
+#W                                                            & Alper Odabas
+#Y  Copyright (C) 2014-2024, Zekeriya Arvasi & Alper Odabas,  
 ##
 
 ## read the actual code 

tst/algebra.tst

@@ -134,7 +134,7 @@ gap> Projection( P, 1 );
   (Z(5)^0)*(1,4)(2,5)(3,6), (Z(5)^0)*(1,5,3)(2,6,4), (Z(5)^0)*(1,6,5,4,3,2), 
   <zero> of ..., <zero> of ... ]
 gap> P3 := SemidirectProductOfAlgebras( Q13, act3, A1 );
-<algebra of dimension 5 over Rationals>
+Q13 |X A1
 gap> Embedding( P3, 1 );
 [ v.1, v.2 ] -> [ v.1, v.2 ]
 gap> Embedding( P3, 2 );

tst/convert.tst

@@ -48,10 +48,9 @@ gap> C3 := Cat1AlgebraSelect( 2, 6, 2, 4 );;
 gap> ############################ 
 gap> ## Chapter 5,  Section 5.1.1
 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:
 
 [ 
@@ -60,11 +59,35 @@ 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 ]
 gap> X3 := XModAlgebraOfCat1Algebra( C3 );
 [ <algebra of dimension 3 over GF(2)> -> <algebra of dimension 3 over GF(2)> ]
 gap> Display( X3 ); 

tst/module3.tst

@@ -104,6 +104,26 @@ Crossed module [A(M3)->A3] :-
   [ [ 0, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ], 
   [ [ 0, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ] ]
 
+## Chapter 5, Section 5.1.1
+gap> C3 := Cat1AlgebraOfXModAlgebra( X3 );
+[A3 |X A(M3)=>A3]
+gap> Display( C3 );           
+Cat1-algebra [A3 |X A(M3)=>A3] :- 
+:  range algebra has generators:
+  [ [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] ]
+: tail homomorphism = head homomorphism
+  they map the source generators to:
+  [ [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ], 
+  [ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ], 
+  [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ], 
+  [ [ 0, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ], 
+  [ [ 0, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ], 
+  [ [ 0, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ] ]
+: range embedding maps range generators to:
+  [ v.1 ]
+: kernel has generators:
+  [ v.4, v.5, v.6 ]
+
 gap> SetInfoLevel( InfoXModAlg, level );; 
 gap> STOP_TEST( "module.tst", 10000 );