remove unused global function ElementsLeftActing

doc/algebra.xml

@@ -2,7 +2,7 @@
 <!--                                                                     -->
 <!--  algebra.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                            --> 
 <!--                                                                     -->
 <!-- ------------------------------------------------------------------- -->
@@ -19,7 +19,8 @@ All the algebras considered in this package will be associative
 and commutative. 
 Scalars belong to a commutative ring <B>k</B> with <M>1 \neq 0</M>. 
 <P/>
-<E>Why not a field?  A group ring over the integers is not an algebra.</E>
+<E>(Why not a field?  
+A group ring over the integers is not an algebra. [CDW])</E>
 <P/>
 
 
@@ -137,7 +138,7 @@ This operation returns <C>MultiplierAlgebraOfIdealBySubalgebra(A,A,A);</C>.
 
 <Example>
 <![CDATA[
-gap> MA5c6 := RegularMultiplierAlgebra( A5c6 );
+gap> MA5c6 := RegularAlgebraMultiplier( A5c6 );
 <algebra of dimension 6 over GF(5)>
 gap> vecM := BasisVectors( Basis( MA5c6 ) );; 
 gap> vecM[3];
@@ -150,9 +151,10 @@ gap> vecM[3];
    <Attr Name="MultiplierHomomorphism"
          Arg="M" />
 <Description>
-If <M>M</M> is a multiplier algebra with elements of algebra <M>A</M> 
+If <M>M</M> is a multiplier algebra with elements of 
+a subalgebra <M>B</M> of an algebra <M>A</M> 
 multiplying an ideal <M>I</M> then this operation returns the 
-homomorphism from <M>A</M> to <M>M</M> mapping <M>a</M> to <M>\mu_a</M>. 
+homomorphism from <M>B</M> to <M>M</M> mapping <M>b</M> to <M>\mu_b</M>. 
 </Description> 
 </ManSection> 
 
@@ -209,10 +211,11 @@ for all <M>k \in </M><B>k</B>, <M>r,r' \in R</M>, and <M>s,s' \in S</M>.
 
 <ManSection>
    <Oper Name="AlgebraActionByMultipliers"
-         Arg="A I" />
+         Arg="A I B" />
 <Description>
-When <M>I</M> is an ideal in <M>A</M> we have seen that the multiplier 
-homomorphism from <M>A</M> to <C>MultiplierAlgebraOf(Ideal(A,I)</C> 
+When <M>I</M> is an ideal in <M>A</M> 
+and <M>B</M> is a subalgebra of <M>A</M>, 
+we have seen that the multiplier homomorphism from <M>A</M> to <C>MultiplierAlgebraOfIdealBySubalgebra(A,I,B)</C> 
 is an action. 
 <P/> 
 In the example the algebra is the group ring of the cyclic group <M>C_6</M> 
@@ -231,7 +234,7 @@ gap> vecA := BasisVectors( Basis( A5c6 ) );;
 gap> v := vecA[1] + vecA[3] + vecA[5];
 (Z(5)^0)*()+(Z(5)^0)*(1,3,5)(2,4,6)+(Z(5)^0)*(1,5,3)(2,6,4)
 gap> I5c6 := Ideal( A5c6, [v] );; 
-gap> actm := AlgebraActionByMultipliers( A5c6, I5c6 );; 
+gap> actm := AlgebraActionByMultipliers( A5c6, I5c6, A5c6 );; 
 gap> actm2 := Image( actm, vecA[2] );; 
 gap> Image( actm2, v );
 (Z(5)^0)*(1,2,3,4,5,6)+(Z(5)^0)*(1,4)(2,5)(3,6)+(Z(5)^0)*(1,6,5,4,3,2)
@@ -242,12 +245,13 @@ gap> Image( actm2, v );
    <Oper Name="AlgebraActionBySurjection"
          Arg="hom" />
 <Description>
-Let <M>\theta : S \to R</M> be a surjective algebra homomorphism 
-such that <M>ks = 0_S ~\forall~ k \in K = \ker\theta</M>. 
-Then <M>R</M> acts on <M>S</M> with <M>r \cdot s = (\theta^{-1}r)s</M>. 
-Note that thus action is well defined since if <M>\theta p = r</M> then 
-<M>\theta^{-1}r = \{ p+k ~|~ k \in \ker\theta \}</M> 
-and <M>(p+k)s = ps+ks = ps+0</M>. 
+Let <M>\theta : A \to B</M> be a surjective algebra homomorphism 
+such that <M>\ker\theta</M> is contained in the annihilator of <M>A</M>. 
+Then <M>B</M> acts on <M>A</M> by <M>b \cdot a = pa</M> 
+where <M>p \in (\theta^{-1}b)</M>. 
+Note that this action is well defined since 
+<M>\theta^{-1}b = \{ p+k ~|~ k \in \ker\theta \}</M> 
+and <M>(p+k)a = pa+ka = ca+0</M>. 
 <P/>
 Continuing with the previous example, 
 we construct the quotient algebra <M>Q5c6 = A5c6/I5c6</M>, 
@@ -276,14 +280,27 @@ gap> m^2;
 gap> m^3;
 [ [ 0, 0, 0, 1 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ] ]
 gap> A1 := Algebra( Rationals, [m] );;
+gap> SetName( A1, "A1" );
 gap> A3 := Subalgebra( A1, [m^3] );; 
 gap> nat3 := NaturalHomomorphismByIdeal( A1, A3 ); 
-<linear mapping by matrix, <algebra of dimension 
-3 over Rationals> -> <algebra of dimension 2 over Rationals>>
+<linear mapping by matrix, A1 -> <algebra of dimension 2 over Ration\
+als>>
+gap> Q13 := Image( nat3 );;
+gap> SetName( Q13, "Q13" );
+gap> Display(nat3);
+LeftModuleHomomorphismByMatrix( Basis( A1, 
+[ [ [ 0, 0, 0, 1 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ] ], 
+  [ [ 0, 1, 2, 3 ], [ 0, 0, 1, 2 ], [ 0, 0, 0, 1 ], [ 0, 0, 0, 0 ] ], 
+  [ [ 0, 0, 1, 4 ], [ 0, 0, 0, 1 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ] ] ] ), 
+[ [ 0, 0 ], [ 1, 0 ], [ 0, 1 ] ], CanonicalBasis( Q13 ) )
 gap> act3 := AlgebraActionBySurjection( nat3 );; 
-gap> a3 := Image( act3, BasisVectors( Basis( Image( nat3 ) ) )[1] );;  
-gap> [ Image( a3, m ) = m^2, Image( a3, m^2 ) = m^3 ];
-[ true, true ]
+gap> B3 := Image( act3 );;
+gap> bvB3 := BasisVectors( Basis( B3 ) );;
+gap> b1 := bvB3[1];;  b2 := bvB3[2];;
+gap> [ Image(b1,m)=m^2, Image(b1,m^2)=m^3, Image(b1,m^3)=Zero(A1) ];
+[ true, true, true ]
+gap> [ Image(b2,m)=m^3, b2=b1^2 ];
+[true, true ]
 ]]>
 </Example>
 
@@ -302,7 +319,7 @@ where the product is given by:
 (r_1,s_1)(r_2,s_2) ~=~ (r_1r_2,~ r_1 \cdot s_2 + r_2 \cdot s_1 + s_1s_2). 
 </Display> 
 
-This product, as wekll as being commutative, is associative: 
+This product, as well as being commutative, is associative: 
 <M>(r_1,s_1)(r_2,s_2)(r_3,s_3)</M> expands as: 
 <Display>
 (r_1r_2r_3,~ \left (r_1r_2)\cdot s3 + (r_1r_3)\cdot s_2 + (r_2r_3)\cdot s_1 
@@ -328,6 +345,7 @@ Continuing the example above,
 <Example>
 <![CDATA[
 gap> P := SemidirectProductOfAlgebras( A5c6, actm, I5c6 ); 
+<algebra of dimension 8 over GF(5)>
 gap> Embedding( P, 1 );
 [ (Z(5)^0)*(), (Z(5)^0)*(1,2,3,4,5,6), (Z(5)^0)*(1,3,5)(2,4,6), 
   (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) 
@@ -341,6 +359,15 @@ gap> Projection( P, 1 );
 [ (Z(5)^0)*(), (Z(5)^0)*(1,2,3,4,5,6), (Z(5)^0)*(1,3,5)(2,4,6), 
   (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>
+gap> Embedding( P3, 1 );
+[ v.1, v.2 ] -> [ v.1, v.2 ]
+gap> Embedding( P3, 2 );
+[ [ [ 0, 1, 2, 3 ], [ 0, 0, 1, 2 ], [ 0, 0, 0, 1 ], [ 0, 0, 0, 0 ] ], 
+  [ [ 0, 0, 1, 4 ], [ 0, 0, 0, 1 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ] ], 
+  [ [ 0, 0, 0, 1 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ] ] ] -> 
+[ v.3, v.4, v.5 ]
 ]]>
 </Example> 
 

doc/bib.xml

@@ -174,7 +174,6 @@
   <title>Higher dimensional algebroids and crossed complexes</title> 
   <school>University of Wales, Bangor (U.K.)</school>  
   <year>1986</year> 
-  <type>Ph.D. thesis</type>
 </phdthesis></entry>
 
 <entry id="aodabas1"><phdthesis> 
@@ -184,7 +183,6 @@
   <title>Crossed Modules of Algebras with GAP</title> 
   <year>2009</year> 
   <school>Osmangazi University, Eskisehir</school> 
-  <type>Ph.D. thesis</type>
 </phdthesis></entry>
 
 <entry id="porter1"><article>

doc/xmod.xml

@@ -78,7 +78,7 @@ gap> B := Subalgebra( A, [m] );;
 gap> SetName( B, "A(m)" ); 
 gap> IsIdeal( A, B ); 
 true
-gap> act := AlgebraActionByMultipliers( A, B );; 
+gap> act := AlgebraActionByMultipliers( A, B, A );; 
 gap> XAB := XModAlgebraByIdeal( A, B ); 
 [ A(m) -> A(l,m) ]
 gap> SetName( XAB, "XAB" ); 
@@ -153,10 +153,10 @@ IdentityMapping( <algebra of dimension 3 over GF(5)> )
          Arg="f" />
 <Description> 
 Let <M>\partial : S\rightarrow R</M> be a surjective algebra homomorphism 
-whose kernel lise in the annihilator of <M>S</M>. 
+whose kernel lies in the annihilator of <M>S</M>. 
 Define the action of <M>R</M> on <M>S</M> by <M>r\cdot s = \widetilde{r}s</M>
 where <M>\widetilde{r} \in \partial^{-1}(r)</M>, 
-as described in section <Ref Oper="AlgebraActionBySurjection"/>
+as described in section <Ref Oper="AlgebraActionBySurjection"/>.
 Then <M>\mathcal{X}=(\partial : S\rightarrow R)</M> 
 is a crossed module with the defined action.
 <P/> 
@@ -170,10 +170,10 @@ Continuing with the example in that section,
 gap> X3 := XModAlgebraBySurjection( nat3 );; 
 gap> Display( X3 ); 
 
-Crossed module [..->..] :- 
-: Source algebra has generators:
+Crossed module [A1->Q13] :- 
+: Source algebra A1 has generators:
   [ [ [ 0, 1, 2, 3 ], [ 0, 0, 1, 2 ], [ 0, 0, 0, 1 ], [ 0, 0, 0, 0 ] ] ]
-: Range algebra has generators:
+: Range algebra Q13 has generators:
   [ v.1, v.2 ]
 : Boundary homomorphism maps source generators to:
   [ v.1 ]
@@ -374,19 +374,19 @@ true
 
 <Section><Heading>(Pre-)Crossed Module Morphisms</Heading>
 
-Let <M>\mathcal{X} = (\partial:S\rightarrow R)</M>, 
+Let <M>\mathcal{X} = (\partial:S\rightarrow R)</M> and 
 <M>\mathcal{X}^{\prime} = 
 (\partial^{\prime }:S^{\prime }\rightarrow R^{\prime })</M> 
-be (pre)crossed modules and <M>\theta :S\rightarrow S^{\prime }</M>, 
+be (pre)crossed modules and let <M>\theta :S\rightarrow S^{\prime }</M>, 
 <M>\varphi : R\rightarrow R^{\prime }</M> be algebra homomorphisms. 
-If
+Then if
 <Display>
 \varphi \circ \partial = \partial ^{\prime } \circ \theta, 
 \qquad 
 \theta (r\cdot s)=\varphi(r) \cdot \theta (s), 
 </Display>
 for all <M>r\in R</M>, <M>s\in S,</M> 
-then the pair <M>(\theta ,\varphi )</M> is called a morphism between 
+the pair <M>(\theta ,\varphi )</M> is called a morphism between 
 <M>\mathcal{X}</M> and <M>\mathcal{X}^{\prime } </M>
 <P/>
 The conditions can be thought as the commutativity of the following diagrams: 

lib/alg2obj.gi

@@ -119,10 +119,12 @@ function( bdy, act )
         fi;
     else
         if not ( B = BB ) then
-            return false;
+            return fail;
         fi;
     fi;
 
+    ## ????  should there be any tests for condition XModAlg1 here ????
+
     PM := rec();
     ObjectifyWithAttributes( PM, NewType( fam, filter ),
         Source, B,
@@ -426,7 +428,7 @@ InstallMethod( XModAlgebraByMultiplierAlgebra,
 function( A )
     local MA, PM, act, bdy; 
     MA := MultiplierAlgebra( A ); 
-    bdy := MultiplierHomomorphism( A );
+    bdy := MultiplierHomomorphism( MA );
     act := IdentityMapping( MA ); 
     SetIsAlgebraAction( act, true ); 
     SetAlgebraActionType( act, "multiplier" ); 

lib/algebra.gd

@@ -28,13 +28,6 @@ DeclareOperation( "InclusionMappingAlgebra", [ IsAlgebra, IsAlgebra ] );
 DeclareOperation( "RestrictionMappingAlgebra", 
    [ IsAlgebraHomomorphism, IsAlgebra, IsAlgebra ] );
 
-## remove this "not IsBound(...)" hack as soon as the declarations 
-## have been moved to the main GAP library
-if not IsBound( AlgebraHomomorphismByFunction ) then
-    DeclareOperation( "AlgebraHomomorphismByFunction", 
-        [ IsAlgebra, IsAlgebra, IsFunction ] );
-fi;
-
 DeclareOperation( "ModuleHomomorphism",
     [ IsAlgebra, IsRing ] ); 
 
@@ -47,20 +40,17 @@ DeclareOperation( "AllIdempotentAlgebraHomomorphisms",
 
 ##############################  algebra actions  #################### 
 
-DeclareGlobalFunction( "ElementsLeftActing" );
-
 DeclareProperty( "IsAlgebraAction", IsMapping );
-
 DeclareGlobalFunction( "AlgebraAction" );
 DeclareAttribute( "LeftElementOfCartesianProduct", IsAlgebraAction );
 DeclareAttribute( "AlgebraActionType", IsAlgebraAction );
 DeclareAttribute( "HasZeroModuleProduct", IsAlgebraAction );
 
+DeclareOperation( "AlgebraActionByMultipliers", 
+    [ IsAlgebra, IsAlgebra, IsAlgebra ] );
 DeclareOperation( "AlgebraAction2", [ IsAlgebra ] );
 DeclareOperation( "AlgebraActionBySurjection", [ IsAlgebraHomomorphism ] );
 DeclareOperation( "AlgebraActionByModule", [ IsAlgebra, IsRing ] );
-DeclareOperation( "AlgebraActionByMultipliers", 
-    [ IsAlgebra, IsAlgebra, IsAlgebra ] );
 
 DeclareOperation ( "SemidirectProductOfAlgebras", 
     [ IsAlgebra, IsAlgebraAction, IsAlgebra ] );