Make congruence poset object a digraph #385

mtorpey · 2017-09-27T12:24:23Z

This probably should've been the case when the object was first created. A poset of semigroup congruences is a digraph with vertices labelled by the congruences themselves. An edge i -> j exists if congruence ~~i~~ j is a subrelation (a refinement) of congruence ~~j~~ i.

This is in reference to Issue #369.

EDIT: i and j reversed.

wilfwilson · 2017-09-27T13:27:06Z

An edge i -> j exists if congruence i is a subrelation (a refinement) of congruence j.

That's not what you've implemented:

gap> S := PartitionMonoid(2);;
gap> lat := LatticeOfCongruences(S);;
gap> univ := UniversalSemigroupCongruence(S);;
gap> triv := SemigroupCongruence(S, []);;
gap> pos_univ := Position(DigraphVertexLabels(lat), univ);;
gap> pos_triv := Position(DigraphVertexLabels(lat), triv);;
gap> IsDigraphEdge(lat, [pos_univ, pos_triv]);
true
gap> IsSubrelation(triv, univ);
false

Which do you want?

mtorpey · 2017-09-27T13:34:16Z

My documentation for IsCongruencePoset says

an edge from vertex i to vertex j if and only if the congruence numbered j is a subrelation of the congruence numbered i.

which is what I intend.

My description in the first comment of this PR is wrong.

wilfwilson · 2017-10-04T14:29:42Z

Thanks for clearing that up @mtorpey. My comments in this post are all about making sure that the behaviour of your posets is consistent with Digraphs.

Here's the documentation:

A congruence poset is a digraph with a vertex for each congruence, and an edge from vertex i to vertex j if and only if the congruence numbered j is a subrelation of the congruence numbered i.

You should make your posets reflexive, since a congruence contains itself. Furthermore, it makes sense to do this for the following reason. Your posets are currently antisymmetric and transitive, but since they're not reflexive, IsPartialOrderDigraph returns false:

gap> S := SymmetricInverseMonoid(3);;
gap> lat := LatticeOfCongruences(S);;
gap> IsAntisymmetricDigraph(lat) and IsTransitiveDigraph(lat);
true
gap> IsReflexiveDigraph(lat) or IsPartialOrderDigraph(lat);
false

I think it's natural to expect a poset of congruences to satisfy IsPartialOrderDigraph. If you make this change, then when a poset of congruences is created, you can set IsPartialOrderDigraph to be true (or you can make this inherent for any poset of congruences).

More fundamentally, you should change the definition of the partial order that is used for your posets. In your setup, x -> y is an edge if y ⊆ x. However this doesn't work well with Digraphs. From the Digraphs documentation:

For a partial order digraph digraph, the corresponding partial order is the relation ≤, defined by x≤y if and only if [x, y] (aka x -> y) is an edge of digraph.

So, taking this all together, cong1 ≤ cong2 if and only if cong2 ⊆ cong1, which I hope you can see is strange. Here's an example of where the problem kicks in:

The join of two elements partially ordered by less-than-or-equal is their least unique upper bound. Therefore, with your congruences, the join of two elements c1 and c2 is greater than both c1 and c2, and is therefore contained in c1 and c2. But (obviously) the join of c1 andc2 contains c1 and c2. This means that, according to Digraphs, a JoinSemilatticeOfCongruences is a meet semilattice rather than a join semilattice.

gap> S := Semigroup([
>  Transformation([1, 3, 3, 3]),
>  Transformation([1, 2, 1])]);;
gap> cong1 := SemigroupCongruence(S,
>  [Transformation([1, 1, 1, 1]), Transformation([1, 2, 1])]);;
gap> cong2 := SemigroupCongruence(S,
>  [Transformation([1, 1, 1, 1]), Transformation([1, 3, 3, 3])]);;
gap> poset := JoinSemilatticeOfCongruences([cong1, cong2],
>   JoinSemigroupCongruences);;
gap> poset := DigraphAddAllLoops(poset);; # make it a partial order
gap> IsPartialOrderDigraph(poset);
true
gap> IsJoinSemilatticeDigraph(poset);
false
gap> IsMeetSemilatticeDigraph(poset);
true

Let me know if you want more convincing about this.

wilfwilson

I've review your PR, and it looks good. Most of my comments are things that will need to change once you've switched the direction of the partial order.

wilfwilson · 2017-10-04T14:43:31Z

gap/congruences/conglatt.gi

  return poset;
 end;

 InstallMethod(MinimalCongruences,
 "for a congruence poset",
 [IsCongruencePoset],
-poset -> CongruencesOfPoset(poset){DigraphSinks(poset!.po)});
+poset -> CongruencesOfPoset(poset){DigraphSinks(poset)});


When you add loops at every vertex, you'll have to change this. Options: vertices with out-degree 1, or vertices with in-degree 1. Or remove loops and then get the sinks.

wilfwilson · 2017-10-04T14:43:42Z

doc/conglatt.xml

-      obtained using <Ref Attr = "CongruencesOfPoset"/>.
+      A congruence poset is a digraph
+      (see <Ref Filt = "IsDigraph" BookName = "Digraphs"/>) with a vertex for
+      each congruence, and an edge from vertex <C>i</C> to vertex <C>j</C> if


wilfwilson · 2017-10-04T14:44:07Z

doc/conglatt.xml

+gap> IsDigraph(poset);
+true
+gap> OutNeighbours(poset);
+[ [  ], [ 1 ], [ 1, 2, 4 ], [ 1, 2 ] ]


This will presumably change.

wilfwilson · 2017-10-04T14:44:48Z

doc/conglatt.xml

 gap> LatticeOfRightCongruences(S);
+<poset of 5 congruences over <regular transformation monoid 
+ of size 3, degree 2 with 2 generators>>
+gap> OutNeighbours(LatticeOfRightCongruences(S));
 [ [  ], [ 1 ], [ 1 ], [ 1 ], [ 1, 2, 3, 4 ] ]


This will presumably change too.

wilfwilson · 2017-10-04T14:45:39Z

gap/congruences/conglatt.gi

-  children := OutNeighboursMutableCopy(poset!.po);
-  parents := InNeighboursMutableCopy(poset!.po);
+  children := OutNeighboursMutableCopy(poset);
+  parents := InNeighboursMutableCopy(poset);


wilfwilson · 2017-10-04T14:46:13Z

gap/congruences/conglatt.gi

-  children := OutNeighboursMutableCopy(poset!.po);
-  parents := InNeighboursMutableCopy(poset!.po);
+  children := OutNeighboursMutableCopy(poset);
+  parents := InNeighboursMutableCopy(poset);


wilfwilson · 2017-10-04T14:46:25Z

gap/congruences/conglatt.gi

-  SetInNeighbours(po, parents);
-  poset := Objectify(NewType(FamilyObj(children), IsCongruencePoset),
-                     rec(po := po));
+  poset := Digraph(children);


wilfwilson · 2017-10-04T14:46:35Z

gap/congruences/conglatt.gi

-  SetInNeighbours(po, parents);
-  poset := Objectify(NewType(FamilyObj(children), IsCongruencePoset),
-                     rec(po := po));
+  poset := Digraph(children);


wilfwilson · 2017-10-04T14:47:58Z

gap/congruences/conglatt.gi

  fi;

-  rel := List([1 .. Length(poset)], x -> Filtered(poset[x], y -> x <> y));
+  rel := List([1 .. Length(out_nbs)], x -> Filtered(out_nbs[x], y -> x <> y));


Change to rel := InNeighbours(DigraphReflexiveTransitiveReduction(poset));

The remove the line j := Difference(rel[i], Union(rel{rel[i]}));.

wilfwilson · 2017-10-04T14:48:15Z

gap/congruences/conglatt.gi

@@ -510,14 +500,15 @@ InstallMethod(DotString,
 "for a congruence poset and a record",
 [IsCongruencePoset, IsRecord],
 function(poset, opts)
-  local congs, S, symbols, i, nr, rel, str, j, k;
+  local out_nbs, congs, S, symbols, i, nr, rel, str, j, k;
+  out_nbs := OutNeighbours(poset);


Dont need this line. Throughout this method, change Length(out_nbs) and Length(rel) to Size(poset).

mtorpey · 2017-10-04T16:19:25Z

Added changes to reverse the ordering.

I still need to add loops.

mtorpey · 2017-10-13T14:24:59Z

This is done!

mtorpey added 3 commits September 27, 2017 12:54

conglatt: make IsCongruenceLattice a digraph subcategory

9769f01

tests: update for congruence lattices as digraphs

1ef2c99

doc: update descriptions of IsCongruenceLattice

b84d8d2

mtorpey added the 3.1 label Sep 27, 2017

wilfwilson requested changes Oct 4, 2017

View reviewed changes

conglatt: reverse ordering so i->j iff i is a subrelation of j

1c81ada

wilfwilson approved these changes Oct 13, 2017

View reviewed changes

conglatt: add loops

c7aace5

mtorpey force-pushed the lattice-digraph branch from e9b6777 to c7aace5 Compare October 13, 2017 14:46

wilfwilson merged commit 42445da into semigroups:master Oct 13, 2017

mtorpey deleted the lattice-digraph branch October 16, 2017 12:36

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Make congruence poset object a digraph #385

Make congruence poset object a digraph #385

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mtorpey commented Oct 13, 2017

Make congruence poset object a digraph #385

Make congruence poset object a digraph #385

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