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semifp: Add a method for IsomorphismFpSemigroup #426

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semifp: Add a method for IsomorphismFpSemigroup #426

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f2b555a
semifp: Add a method for IsomorphismFpSemigroup
Dec 20, 2017
c978387
semifp: fix to IsomorphismFpMonoid
Dec 20, 2017
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98 changes: 97 additions & 1 deletion gap/semigroups/semifp.gi
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Original file line number Diff line number Diff line change
Expand Up @@ -298,7 +298,7 @@ function(S)
spos := Position(sgens, sgens[i]);
mpos := Position(mgens, sgens[i], start[spos]);
lookup[i] := mpos;
if spos <> fail then
if mpos <> fail then
start[spos] := mpos;
fi;
od;
Expand Down Expand Up @@ -541,3 +541,99 @@ IsCollsElms, [IsFpSemigroup, IsElementOfFpSemigroup],
function(S, x)
return SEMIGROUPS.ExtRepObjToWord(ExtRepOfObj(x));
end);

# This method is based on the following paper
# Presentations of Factorizable Inverse Monoids
# David Easdown, James East, and D. G. FitzGerald
# July 19, 2004
InstallMethod(IsomorphismFpSemigroup,
"for an inverse partial perm semigroup",
[IsPartialPermSemigroup and IsInverseActingSemigroupRep],
function(M)
local add_to_odd_positions, S, SS, G, GG, s, g, F, rels, fam, alpha,
beta, lhs, rhs, map, H, o, comp, U, rhs_list, conj, MF, T, inv, rel, x, y, m,
i;

if not IsFactorisableInverseMonoid(M) then
TryNextMethod();
fi;

add_to_odd_positions := function(list, s)
list{[1, 3 .. Length(list) - 1]} := list{[1, 3 .. Length(list) - 1]} + s;
return list;
end;

S := Semigroup(IdempotentGeneratedSubsemigroup(M));
SS := GeneratorsOfSemigroup(S);
G := GroupOfUnits(M);
GG := GeneratorsOfSemigroup(G);
s := Length(GeneratorsOfSemigroup(S));
g := Length(GeneratorsOfSemigroup(G));

F := FreeSemigroup(s + g);
rels := [];
fam := ElementsFamily(FamilyObj(F));

# R_S - semigroup relations for the idempotent generated subsemigroup
alpha := IsomorphismFpSemigroup(S);
for rel in RelationsOfFpSemigroup(Image(alpha)) do
Add(rels, [ObjByExtRep(fam, ExtRepOfObj(rel[1])),
ObjByExtRep(fam, ExtRepOfObj(rel[2]))]);
od;

# R_G - semigroup relations for the group of units
beta := IsomorphismFpSemigroup(G);
for rel in RelationsOfFpSemigroup(Image(beta)) do
lhs := add_to_odd_positions(ShallowCopy(ExtRepOfObj(rel[1])), s);
rhs := add_to_odd_positions(ShallowCopy(ExtRepOfObj(rel[2])), s);
Add(rels, [ObjByExtRep(fam, lhs), ObjByExtRep(fam, rhs)]);
od;

# R_product - see page 4 of the paper
for x in [1 .. s] do
for y in [1 .. g] do
rhs := Factorization(S, SS[x] ^ (GG[y] ^ -1));
Add(rels, [F.(s + y) * F.(x),
EvaluateWord(GeneratorsOfSemigroup(F), rhs) * F.(s + y)]);
od;
od;

map := InverseGeneralMapping(IsomorphismPermGroup(G));
H := Source(map);
o := Enumerate(LambdaOrb(M));
#R_tilde - see page 4 of the paper
for m in [2 .. Length(OrbSCC(o))] do
comp := OrbSCC(o)[m];
U := SmallGeneratingSet(Stabilizer(H,
PartialPerm(o[comp[1]], o[comp[1]]),
OnRight));

for i in comp do
rhs_list := Factorization(S, PartialPerm(o[i], o[i]));
rhs := EvaluateWord(GeneratorsOfSemigroup(F), rhs_list);
conj := MappingPermListList(o[comp[1]], o[i]);
for x in List(U, x -> x ^ conj) do
lhs := ShallowCopy(rhs_list);
Append(lhs, Factorization(G, x ^ map) + s);
Add(rels, [EvaluateWord(GeneratorsOfSemigroup(F), lhs), rhs]);
od;
od;
od;

# Relation to indentify One(G) and One(S)
Add(rels, [EvaluateWord(GeneratorsOfSemigroup(F),
Factorization(G, One(G)) + s),
EvaluateWord(GeneratorsOfSemigroup(F),
Factorization(S, One(S)))]);

MF := F / rels; # FpSemigroup which is isomorphic to M, with different gens.
fam := ElementsFamily(FamilyObj(MF));
T := Semigroup(Concatenation(SS, GG)); # M with isomorphic generators to MF

map := x -> ElementOfFpSemigroup(fam, EvaluateWord(GeneratorsOfSemigroup(F),
Factorization(T, x)));
inv := x -> EvaluateWord(GeneratorsOfSemigroup(T),
SEMIGROUPS.ExtRepObjToWord(ExtRepOfObj(x)));

return MagmaIsomorphismByFunctionsNC(M, MF, map, inv);
end);
61 changes: 61 additions & 0 deletions tst/standard/semifp.tst
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Original file line number Diff line number Diff line change
Expand Up @@ -2060,6 +2060,67 @@ gap> SEMIGROUPS.ExtRepObjToString([100, 1]);
Error, SEMIGROUPS.ExtRepObjToString: the maximum value in an odd position of t\
he argument must be at most 52,

# Test IsomorphismFpSemigroup (for factorizable inverse monoids)
gap> S := SymmetricInverseMonoid(4);;
gap> iso := IsomorphismFpSemigroup(S);;
gap> BruteForceIsoCheck(iso);
true
gap> BruteForceInverseCheck(iso);
true
gap> S := InverseSemigroup(
> [PartialPerm([1, 2, 3, 4, 5, 6, 7, 8], [2, 4, 8, 6, 3, 1, 5, 7]),
> PartialPerm([1, 2, 3, 4, 5, 6, 7, 8], [3, 5, 4, 7, 6, 8, 1, 2]),
> PartialPerm([1, 2, 3, 4, 5, 6, 7, 8], [4, 6, 7, 1, 8, 2, 3, 5]),
> PartialPerm([], [])]);;
gap> iso := IsomorphismFpSemigroup(S);;
gap> BruteForceIsoCheck(iso);
true
gap> BruteForceInverseCheck(iso);
true
gap> S := InverseSemigroup(
> [PartialPerm([1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [9, 10, 8, 2, 1, 7, 5, 4, 6, 3]),
> PartialPerm([1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [10, 9, 3, 6, 5, 2, 8, 1, 4, 7]),
> PartialPerm([], [])]);;
gap> iso := IsomorphismFpSemigroup(S);;
gap> BruteForceIsoCheck(iso);
true
gap> BruteForceInverseCheck(iso);
true
gap> tst := [InverseMonoid([PartialPerm([1, 2, 3, 4, 5], [4, 5, 2, 3, 1]),
> PartialPerm([1, 3], [1, 3])]),
> InverseMonoid([PartialPerm([1, 2, 3, 4], [1, 2, 3, 4]),
> PartialPerm([1, 2, 3, 4, 5], [3, 1, 5, 4, 2])]),
> InverseMonoid([PartialPerm([1, 2, 3, 4, 5], [5, 4, 2, 3, 1]),
> PartialPerm([1, 2, 4], [1, 2, 4])]),
> InverseMonoid([PartialPerm([1, 2, 5], [2, 1, 5]),
> PartialPerm([1, 2], [1, 2])]),
> InverseMonoid([PartialPerm([1, 2, 3], [1, 4, 5]),
> PartialPerm([1, 2, 3, 4, 5], [1, 5, 4, 2, 3])]),
> InverseMonoid([PartialPerm([1, 2, 3, 4, 5], [3, 1, 5, 4, 2]),
> PartialPerm([1, 2, 3, 4, 5], [5, 1, 3, 4, 2])]),
> InverseMonoid([PartialPerm([1, 2, 5], [2, 3, 5]),
> PartialPerm([1, 2, 3, 5], [2, 3, 1, 5])]),
> InverseMonoid([PartialPerm([1, 2, 3, 4, 5], [4, 2, 3, 1, 5]),
> PartialPerm([1, 2, 3, 4, 5], [5, 3, 2, 1, 4])]),
> InverseMonoid([PartialPerm([1, 2, 3, 5], [2, 1, 3, 5]),
> PartialPerm([1, 2, 3, 5], [5, 2, 1, 3])]),
> InverseMonoid([PartialPerm([1, 2, 3], [5, 4, 1]),
> PartialPerm([1, 2, 3, 4, 5], [2, 3, 5, 1, 4])]),
> InverseMonoid([PartialPerm([1, 2, 3, 4, 5], [4, 3, 5, 2, 1]),
> PartialPerm([1, 2, 4, 5], [5, 4, 2, 1]),
> PartialPerm([1, 4], [3, 2]), PartialPerm([1, 2, 3, 4, 5], [2, 3, 5, 1, 4]),
> PartialPerm([1, 2, 5], [2, 3, 4])]),
> InverseMonoid([PartialPerm([1, 2, 3, 4, 5], [2, 4, 1, 5, 3]),
> PartialPerm([1, 3, 4], [2, 1, 3]),
> PartialPerm([1, 2, 3, 4, 5], [4, 1, 2, 5, 3]), PartialPerm([1, 3], [5, 4]),
> PartialPerm([1, 3, 5], [2, 4, 1])])];;
gap> ForAll(tst, IsFactorisableInverseMonoid);
true
gap> ForAll(tst, S -> BruteForceIsoCheck(IsomorphismFpSemigroup(S)));
true
gap> ForAll(tst{[1 .. 10]}, S -> BruteForceInverseCheck(IsomorphismFpSemigroup(S)));
true

#T# SEMIGROUPS_UnbindVariables
gap> Unbind(BruteForceInverseCheck);
gap> Unbind(BruteForceIsoCheck);
Expand Down