Introduce associativity check to AsSemigroup

See #4030.

Fixes #4030.

lib/semigrp.gd

@@ -174,8 +174,9 @@ DeclareOperation( "SemigroupByGenerators", [ IsCollection ] );
 ##  <Oper Name="AsSemigroup" Arg='C'/>
 ##
 ##  <Description>
-##  If <A>C</A> is a collection whose elements form a semigroup
-##  (see&nbsp;<Ref Filt="IsSemigroup"/>) then <Ref Oper="AsSemigroup"/>
+##  If <A>C</A> is a collection whose elements can be multiplied via
+##  <Ref Oper="\*"/> and thereby forms a semigroup
+##  (see&nbsp;<Ref Filt="IsSemigroup"/>), then <Ref Oper="AsSemigroup"/>
 ##  returns this semigroup.
 ##  Otherwise <K>fail</K> is returned.
 ##  </Description>

lib/semigrp.gi

@@ -544,6 +544,10 @@ InstallMethod( AsSemigroup,
     function ( D )
     local   S,  L;
 
+    if not IsAssociative( D ) then
+      return fail;
+    fi;
+
     D := AsSSortedList( D );
     L := ShallowCopy( D );
     S := Submagma( SemigroupByGenerators( D ), [] );

tst/testbugfix/2021-04-08-non-associative-semigroup.tst

@@ -0,0 +1,52 @@
+# https://github.com/gap-system/gap/issues/4030
+#
+gap> T := [
+>   [ 1, 2, 6, 5, 4, 3 ],
+>   [ 2, 1, 4, 2, 2, 5 ],
+>   [ 6, 5, 1, 6, 6, 4 ],
+>   [ 5, 6, 3, 4, 1, 2 ],
+>   [ 4, 3, 2, 1, 5, 6 ],
+>   [ 3, 4, 5, 3, 3, 1 ]
+> ];;
+gap> M := MagmaByMultiplicationTable(T);
+<magma with 6 generators>
+gap> IsAssociative(M);
+false
+gap> AsSemigroup(M);
+fail
+gap> M := MagmaByMultiplicationTable(T);
+<magma with 6 generators>
+gap> AsSemigroup(M);
+fail
+gap> AsSemigroup(Elements(M));
+fail
+
+#
+gap> T := [
+>   [ 1, 2, 3, 4, 5 ],
+>   [ 2, 2, 2, 5, 5 ],
+>   [ 3, 2, 3, 4, 5 ],
+>   [ 4, 2, 4, 3, 5 ],
+>   [ 5, 2, 5, 2, 5 ]
+> ];;
+gap> M := MagmaByMultiplicationTable(T);
+<magma with 5 generators>
+gap> IsAssociative(M);
+true
+gap> S := AsSemigroup(M);;
+gap> IsSemigroup(S);
+true
+gap> Size(S);
+5
+gap> M := MagmaByMultiplicationTable(T);
+<magma with 5 generators>
+gap> S := AsSemigroup(M);;
+gap> IsSemigroup(S);
+true
+gap> Size(S);
+5
+gap> S := AsSemigroup(Elements(M));;
+gap> IsSemigroup(S);
+true
+gap> Size(S);
+5

tst/testinstall/magma.tst

@@ -1,4 +1,4 @@
-#@local M,T
+#@local F,M,T
 gap> START_TEST( "magma.tst" );
 
 #
@@ -35,6 +35,10 @@ gap> IsAssociative(M) or IsCommutative(M);
 false
 gap> Filtered(Combinations(Elements(M)), x -> Size(x) > 0 and IsAssociative(x));
 [ [ m1 ], [ m1, m3 ], [ m2 ], [ m2, m4 ], [ m3 ], [ m4 ] ]
+gap> AsSemigroup([Elements(M)[1], Elements(M)[2]]);
+fail
+gap> AsSemigroup([Elements(M)[1], Elements(M)[3]]);
+<semigroup of size 2, with 2 generators>
 gap> Filtered(Combinations(Elements(M)), x -> Size(x) > 0 and IsCommutative(x));
 [ [ m1 ], [ m1, m2 ], [ m1, m2, m3 ], [ m1, m2, m5 ], [ m1, m3 ], [ m1, m5 ], 
   [ m2 ], [ m2, m3 ], [ m2, m4 ], [ m2, m5 ], [ m3 ], [ m4 ], [ m5 ] ]
@@ -47,6 +51,8 @@ gap> IsCommutative( F );
 false
 gap> Number( Combinations( F, 3 ), IsCommutative );
 1
+gap> AsSemigroup( F );
+<semigroup of size 6, with 2 generators>
 
 #
 gap> STOP_TEST( "magma.tst" );