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Implement IsJoinIrreducible and IsMeetIrreducible #910

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46836b9
Naive implementation of IsJoinIrreducible and IsMeetIrreducible
ThatOtherAndrew Mar 20, 2026
3bb2bb8
Optimise join/meet irreducible functions
ThatOtherAndrew Mar 20, 2026
7b7d883
Add error handling inspired by existing .gi files
ThatOtherAndrew Mar 20, 2026
6c07453
Expand tests with LLM assistance
ThatOtherAndrew Mar 20, 2026
8cf99ca
Integrate joinmeetirreducible into oper.gi/oper.gd
ThatOtherAndrew Mar 20, 2026
2c37a4b
Remove dirty whitespace from tst/standard/oper.tst
ThatOtherAndrew Mar 20, 2026
8ae6cba
Add joinmeetirreducible tests to tst/standard/oper.tst
ThatOtherAndrew Mar 20, 2026
d2fd4fe
Apply suggestion from @james-d-mitchell
ThatOtherAndrew Apr 1, 2026
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3 changes: 3 additions & 0 deletions gap/oper.gd
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Original file line number Diff line number Diff line change
Expand Up @@ -161,3 +161,6 @@ DeclareOperation("PartialOrderDigraphJoinOfVertices",
[IsDigraph, IsPosInt, IsPosInt]);
DeclareOperation("PartialOrderDigraphMeetOfVertices",
[IsDigraph, IsPosInt, IsPosInt]);

DeclareOperation("IsJoinIrreducible", [IsDigraph, IsPosInt]);
DeclareOperation("IsMeetIrreducible", [IsDigraph, IsPosInt]);
32 changes: 32 additions & 0 deletions gap/oper.gi
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Expand Up @@ -2720,6 +2720,38 @@ function(D, i, j)
return fail;
end);

InstallMethod(IsJoinIrreducible,
"for a digraph and a positive integer",
[IsDigraph, IsPosInt],
function(D, v)
local hasse;
if not IsPartialOrderDigraph(D) then
ErrorNoReturn("the 1st argument <D> must satisfy IsPartialOrderDigraph,");
elif not v in DigraphVertices(D) then
ErrorNoReturn("the 2nd argument <v> must be a vertex of the ",
"1st argument <D>,");
fi;
hasse := DigraphReflexiveTransitiveReduction(DigraphMutableCopyIfMutable(D));
# join-irreducible iff at most one lower cover in the Hasse diagram
Comment on lines +2728 to +2735

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The only explicit precondition checked is IsPartialOrderDigraph(D), but IsPartialOrderDigraph does not exclude multidigraphs. If D has multiple edges, DigraphReflexiveTransitiveReduction will throw a different error (“must be a digraph with no multiple edges”) from inside this method. Consider either rejecting multidigraphs up front with a clear error, or normalizing the copied digraph by removing multiple edges before computing the reduction.

Copilot uses AI. Check for mistakes.
return InDegreeOfVertexNC(hasse, v) <= 1;
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end);

InstallMethod(IsMeetIrreducible,
"for a digraph and a positive integer",
[IsDigraph, IsPosInt],
function(D, v)
local hasse;
if not IsPartialOrderDigraph(D) then
ErrorNoReturn("the 1st argument <D> must satisfy IsPartialOrderDigraph,");
elif not v in DigraphVertices(D) then
ErrorNoReturn("the 2nd argument <v> must be a vertex of the ",
"1st argument <D>,");
fi;
hasse := DigraphReflexiveTransitiveReduction(DigraphMutableCopyIfMutable(D));
# meet-irreducible iff at most one upper cover in the Hasse diagram
return OutDegreeOfVertexNC(hasse, v) <= 1;
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Comment on lines +2744 to +2752

Copilot AI Apr 1, 2026

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IsMeetIrreducible relies on “at most one upper cover in the Hasse diagram”, which (like the join case) is only equivalent to “not a meet of two distinct larger elements” when meets exist appropriately (e.g. meet-semilattices/lattices). For arbitrary partial orders, vertices can have multiple upper covers while no such meet representation exists. Either restrict the accepted digraphs to IsMeetSemilatticeDigraph/IsLatticeDigraph, or compute using the actual meet operation to match the documented semantics.

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end);

InstallMethod(DigraphKings, "for a digraph and a positive integer",
[IsDigraph, IsPosInt],
function(D, n)
Expand Down
121 changes: 112 additions & 9 deletions tst/standard/oper.tst
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Expand Up @@ -17,6 +17,7 @@
#@local out, out1, out2, out3, p1, p2, path, preorder, qr, r, res, rtclosure, t
#@local tclosure, u1, u2, x
#@local p, q, idp, idt, M
#@local b2, chain, lattice, m3
gap> START_TEST("Digraphs package: standard/oper.tst");
gap> LoadPackage("digraphs", false);;

Expand Down Expand Up @@ -1514,18 +1515,18 @@ gap> gr := ChainDigraph(5);
<immutable chain digraph with 5 vertices>
gap> DigraphRandomWalk(gr, 2, 100);
[ [ 2, 3, 4, 5 ], [ 1, 1, 1 ] ]
gap> DigraphRandomWalk(gr, 2, 2);
gap> DigraphRandomWalk(gr, 2, 2);
[ [ 2, 3, 4 ], [ 1, 1 ] ]
gap> DigraphRandomWalk(gr, 5, 100);
[ [ 5 ], [ ] ]
gap> gr := CompleteBipartiteDigraph(10, 8);;
gap> DigraphRandomWalk(gr, 3, 0);
gap> DigraphRandomWalk(gr, 3, 0);
[ [ 3 ], [ ] ]
gap> DigraphRandomWalk(gr, 19, 5);
Error, the 2nd argument <v> must be a vertex of the 1st argument <D>,
gap> DigraphRandomWalk(gr, 123, 5);
Error, the 2nd argument <v> must be a vertex of the 1st argument <D>,
gap> DigraphRandomWalk(gr, 3, -1);
gap> DigraphRandomWalk(gr, 3, -1);
Error, the 3rd argument <t> must be a non-negative int,

# DigraphLayers
Expand Down Expand Up @@ -1822,6 +1823,108 @@ fail
gap> PartialOrderDigraphJoinOfVertices(gr1, 3, 4);
fail

# IsJoinIrreducible, IsMeetIrreducible

# The pentagon lattice N5: 1 < {2, 3}, 3 < 4, {2, 4} < 5
gap> gr := Digraph([[2, 3], [5], [4], [5], []]);
<immutable digraph with 5 vertices, 5 edges>
gap> lattice := DigraphReflexiveTransitiveClosure(gr);
<immutable preorder digraph with 5 vertices, 13 edges>
gap> IsJoinIrreducible(lattice, 1);
true
gap> IsJoinIrreducible(lattice, 2);
true
gap> IsJoinIrreducible(lattice, 3);
true
gap> IsJoinIrreducible(lattice, 4);
true
gap> IsJoinIrreducible(lattice, 5);
false
gap> IsMeetIrreducible(lattice, 1);
false
gap> IsMeetIrreducible(lattice, 2);
true
gap> IsMeetIrreducible(lattice, 3);
true
gap> IsMeetIrreducible(lattice, 4);
true
gap> IsMeetIrreducible(lattice, 5);
true

# A 4-element chain: every element is doubly irreducible
gap> chain := DigraphReflexiveTransitiveClosure(ChainDigraph(4));
<immutable preorder digraph with 4 vertices, 10 edges>
gap> IsJoinIrreducible(chain, 1);
true
gap> IsJoinIrreducible(chain, 2);
true
gap> IsJoinIrreducible(chain, 3);
true
gap> IsJoinIrreducible(chain, 4);
true
gap> IsMeetIrreducible(chain, 1);
true
gap> IsMeetIrreducible(chain, 2);
true
gap> IsMeetIrreducible(chain, 3);
true
gap> IsMeetIrreducible(chain, 4);
true

# The Boolean lattice B2: 1 < {2, 3} < 4
gap> b2 := DigraphReflexiveTransitiveClosure(Digraph([[2, 3], [4], [4], []]));
<immutable preorder digraph with 4 vertices, 9 edges>
gap> IsJoinIrreducible(b2, 1);
true
gap> IsJoinIrreducible(b2, 2);
true
gap> IsJoinIrreducible(b2, 3);
true
gap> IsJoinIrreducible(b2, 4);
false
gap> IsMeetIrreducible(b2, 1);
false
gap> IsMeetIrreducible(b2, 2);
true
gap> IsMeetIrreducible(b2, 3);
true
gap> IsMeetIrreducible(b2, 4);
true

# The M3 lattice: 1 < {2, 3, 4} < 5
gap> m3 := DigraphReflexiveTransitiveClosure(Digraph([[2, 3, 4], [5], [5], [5], []]));
<immutable preorder digraph with 5 vertices, 12 edges>
gap> IsJoinIrreducible(m3, 1);
true
gap> IsJoinIrreducible(m3, 2);
true
gap> IsJoinIrreducible(m3, 3);
true
gap> IsJoinIrreducible(m3, 4);
true
gap> IsJoinIrreducible(m3, 5);
false
gap> IsMeetIrreducible(m3, 1);
false
gap> IsMeetIrreducible(m3, 2);
true
gap> IsMeetIrreducible(m3, 3);
true
gap> IsMeetIrreducible(m3, 4);
true
gap> IsMeetIrreducible(m3, 5);
true

# Error handling
gap> IsJoinIrreducible(CycleDigraph(3), 1);
Error, the 1st argument <D> must satisfy IsPartialOrderDigraph,
gap> IsMeetIrreducible(CycleDigraph(3), 1);
Error, the 1st argument <D> must satisfy IsPartialOrderDigraph,
gap> IsJoinIrreducible(Digraph([[1]]), 2);
Error, the 2nd argument <v> must be a vertex of the 1st argument <D>,
gap> IsMeetIrreducible(Digraph([[1]]), 2);
Error, the 2nd argument <v> must be a vertex of the 1st argument <D>,

# DigraphClosure
gap> gr := Digraph([[4, 5, 6, 7, 9], [7, 3], [2, 6, 7, 9, 10],
> [5, 6, 7, 1, 9], [1, 4, 6, 7], [7, 1, 3, 4, 5],
Expand Down Expand Up @@ -2460,9 +2563,9 @@ gap> ConormalProduct(CycleDigraph(2), CycleDigraph(8));
#HomomorphicProduct
gap> D := Digraph([[2, 3], [1, 3, 3], [1, 2, 2]]);
<immutable multidigraph with 3 vertices, 8 edges>
gap> HomomorphicProduct(D, D);
gap> HomomorphicProduct(D, D);
Error, the 1st argument (a digraph) must not satisfy IsMultiDigraph
gap> DigraphSymmetricClosure(CycleDigraph(6));
gap> DigraphSymmetricClosure(CycleDigraph(6));
<immutable symmetric digraph with 6 vertices, 12 edges>
gap> HomomorphicProduct(PetersenGraph(), last);
<immutable digraph with 60 vertices, 1080 edges>
Expand Down Expand Up @@ -2525,7 +2628,7 @@ gap> StrongProduct(NullDigraph(0), CompleteDigraph(3));
<immutable empty digraph with 0 vertices>
gap> D1 := Digraph([[2], [1, 3, 4], [2, 5], [2, 5], [3, 4]]);
<immutable digraph with 5 vertices, 10 edges>
gap> D2 := Digraph([[2], [1, 3, 4], [2], [2]]);
gap> D2 := Digraph([[2], [1, 3, 4], [2], [2]]);
<immutable digraph with 4 vertices, 6 edges>
gap> LexicographicProduct(D1, D2);
<immutable digraph with 20 vertices, 190 edges>
Expand All @@ -2545,7 +2648,7 @@ gap> OutNeighbours(last);
[ 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20 ],
[ 9, 10, 11, 12, 13, 14, 15, 16, 18 ],
[ 9, 10, 11, 12, 13, 14, 15, 16, 18 ] ]
gap> LexicographicProduct(ChainDigraph(3), CycleDigraph(7));
gap> LexicographicProduct(ChainDigraph(3), CycleDigraph(7));
<immutable digraph with 21 vertices, 119 edges>

# SwapDigraphs
Expand Down Expand Up @@ -3032,7 +3135,7 @@ Error, expected an edge between the 2nd and 3rd arguments (vertices) 1 and

# DigraphContractEdge: Edge is a looped edge (u = v)
gap> D := DigraphByEdges([[1, 1], [2, 1], [1, 2]]);;
gap> DigraphVertexLabels(D);;
gap> DigraphVertexLabels(D);;
gap> C := DigraphContractEdge(D, 1, 1);
Error, The 2nd argument <u> must not be equal to the 3rd argument <v>
gap> DigraphHasLoops(D);
Expand Down Expand Up @@ -3196,7 +3299,7 @@ Error, expected an edge between the 2nd and 3rd arguments (vertices) 1 and

# DigraphContractEdge: Edge is a looped edge (u = v) (mutable)
gap> D := DigraphByEdges(IsMutableDigraph, [[1, 1], [2, 1], [1, 2]]);;
gap> DigraphVertexLabels(D);;
gap> DigraphVertexLabels(D);;
gap> DigraphContractEdge(D, 1, 1);
Error, The 2nd argument <u> must not be equal to the 3rd argument <v>
gap> DigraphHasLoops(D);
Expand Down
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