Skip to content

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

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

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

Show file tree
Hide file tree
Changes from all commits
Commits
Show all changes
6 commits
Select commit Hold shift + click to select a range
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
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
35 changes: 25 additions & 10 deletions gap/semigroups/semigraph.gi
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -35,13 +35,18 @@ end);
InstallMethod(AssignGeneratorVariables, "for an inverse semigroup",
[IsGraphInverseSemigroup],
function(S)
DoAssignGenVars(GeneratorsOfInverseSemigroup(S));
# the zero element uses the char 0, which cannot be assigned as a variable
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",
[IsDigraph],
function(graph)
local fam, S, gens, i;
local fam, S, gens, inv_gens, i;

fam := NewFamily("GraphInverseSemigroupElementsFamily",
IsGraphInverseSemigroupElement,
Expand All @@ -66,15 +71,24 @@ function(graph)
fam!.semigroup := S;

gens := [];
inv_gens := [];
SetGraphOfGraphInverseSemigroup(S, graph);

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

gens := Concatenation(inv_gens, List([1 .. DigraphNrEdges(graph)],
x -> inv_gens[x] ^ -1));

# if the graph has 0 or 1 vertex, then the zero element is not generated by
# products of vertices and edges, so must be added to both generating sets
if DigraphNrVertices(graph) < 2 then
Add(gens, MultiplicativeZero(S));
Add(inv_gens, MultiplicativeZero(S));
fi;
SetGeneratorsOfSemigroup(S, gens);
SetGeneratorsOfInverseSemigroup(S, inv_gens);
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
View file
Open in desktop
Original file line number Diff line number Diff line change
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
View file
Open in desktop
Original file line number Diff line number Diff line change
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
Loading