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Adding the zero element to the generating set for graph inverse semigroups with at most one vertex. #1216

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Merged
james-d-mitchell merged 6 commits into from
Sep 6, 2026
Merged

Adding the zero element to the generating set for graph inverse semigroups with at most one vertex. #1216

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852754a
pausing to work on something else
joe-dw Sep 2, 2026
502ec98
pausing to work on something else
joe-dw Sep 2, 2026
8aa0a97
Merge branch 'GIS-with-one-vertex' of https://github.com/joe-dw/semig…
joe-dw Sep 3, 2026
c348b7e
Switched digraph to empty digraph for group and Rees matrix isomorphi…
joe-dw Sep 3, 2026
4b3eaf4
Fix: GraphInverseSemigroup(Digraph([])) has no vertices
joe-dw Sep 3, 2026
014ea36
Refactoring, comments and lint
joe-dw Sep 3, 2026
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23 changes: 19 additions & 4 deletions gap/semigroups/semigraph.gi
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Original file line number Diff line number Diff line change
Expand Up @@ -35,7 +35,11 @@ end);
InstallMethod(AssignGeneratorVariables, "for an inverse semigroup",
[IsGraphInverseSemigroup],
function(S)
DoAssignGenVars(GeneratorsOfInverseSemigroup(S));
if DigraphNrVertices(GraphOfGraphInverseSemigroup(S)) < 2 then
Info(InfoWarning, 1, "zero element is a generator but was not assigned!");
fi;
DoAssignGenVars(Difference(GeneratorsOfInverseSemigroup(S),
[MultiplicativeZero(S)]));
end);

InstallMethod(GraphInverseSemigroup, "for a digraph",
Expand Down Expand Up @@ -66,15 +70,25 @@ function(graph)
fam!.semigroup := S;

gens := [];
SetGraphOfGraphInverseSemigroup(S, graph);
for i in [1 .. DigraphNrVertices(graph) + DigraphNrEdges(graph)] do
Add(gens, Objectify(fam!.type, [[i], graph]));
od;
SetGeneratorsOfSemigroup(S,

if DigraphNrVertices(graph) < 2 then
SetGeneratorsOfSemigroup(S,
Concatenation(gens,
List([1 .. DigraphNrEdges(graph)],
x -> gens[x] ^ -1),
[MultiplicativeZero(S)]));
Add(gens, MultiplicativeZero(S));
else
SetGeneratorsOfSemigroup(S,
Concatenation(gens,
List([1 .. DigraphNrEdges(graph)],
x -> gens[x] ^ -1)));
fi;
SetGeneratorsOfInverseSemigroup(S, gens);
SetGraphOfGraphInverseSemigroup(S, graph);
Comment on lines -72 to -77

@james-d-mitchell james-d-mitchell Sep 3, 2026

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This'd be more compact as (a properly formatted, and working version of):

  SetGraphOfGraphInverseSemigroup(S, graph);
  inv_gens := [];
  for i in [1 .. DigraphNrVertices(graph) + DigraphNrEdges(graph)] do
    Add(inv_gens, Objectify(fam!.type, [[i], graph]));
  od;
  gens := Concatenation(inv_gens, List([1 .. DigraphNrEdges(graph)], x -> gens[x] ^ -1));
  if DigraphNrVertices(graph) < 2 then
     Add(gens, MultiplicativeZero(S));
     Add(inv_gens, MultiplicativeZero(S));
  fi;
  SetGeneratorsOfInverseSemigroup(inv_gens);
  SetGeneratorsOfSemigroup(gens);

if the order of the generators is not important, then you could conditionally add the zero to inv_gens before populating gens. I haven't checked that this actually works.

@joe-dw joe-dw Sep 3, 2026

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I've added it at the end as there are test cases that use the fact that, for any i less than the number of edges,

S.i = e_i

And this way it shouldn't break anything

@james-d-mitchell james-d-mitchell Sep 3, 2026

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I think that makes sense

@joe-dw joe-dw Sep 3, 2026

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Will implement your changes and add some comments

return S;
end);

Expand Down Expand Up @@ -286,7 +300,8 @@ InstallMethod(EdgesOfGraphInverseSemigroup,
"for a graph inverse semigroup",
[IsGraphInverseSemigroup],
S -> Difference(GeneratorsOfInverseSemigroup(S),
VerticesOfGraphInverseSemigroup(S)));
Concatenation([MultiplicativeZero(S)],
VerticesOfGraphInverseSemigroup(S))));

InstallMethod(IndexOfVertexOfGraphInverseSemigroup,
"for a graph inverse semigroup element",
Expand Down
4 changes: 2 additions & 2 deletions tst/standard/semigroups/grpperm.tst
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Expand Up @@ -374,8 +374,8 @@ gap> iso := IsomorphismPermGroup(S);
Error, the argument (a semigroup) does not satisfy IsGroupAsSemigroup

# IsomorphismPermGroup: for a Integer Matrix Semigroup
gap> S := GraphInverseSemigroup(Digraph([[]]));
<finite graph inverse semigroup with 1 vertex, 0 edges>
gap> S := GraphInverseSemigroup(Digraph([]));
<finite graph inverse semigroup with 0 vertices, 0 edges>
gap> iso := IsomorphismPermGroup(S);;
gap> BruteForceIsoCheck(iso); BruteForceInverseCheck(iso);
true
Expand Down
4 changes: 2 additions & 2 deletions tst/standard/semigroups/semirms.tst
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Expand Up @@ -1912,8 +1912,8 @@ true

# AsSemigroup:
# convert from IsGraphInverseSemigroup to IsReesMatrixSemigroup
gap> S := GraphInverseSemigroup(Digraph([[]]));
<finite graph inverse semigroup with 1 vertex, 0 edges>
gap> S := GraphInverseSemigroup(Digraph([]));
<finite graph inverse semigroup with 0 vertices, 0 edges>
gap> T := AsSemigroup(IsReesMatrixSemigroup, S);
<Rees matrix semigroup 1x1 over Group(())>
gap> Size(S) = Size(T);
Expand Down
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