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Decomposition of some Completely Regular Semigroups into Strong Semilattices of Semigroups #731
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cff1a6a
Add IsomorphismSemigroup from Clifford smgp to SSS
tomcontileslie 79d230f
Add IsOrthoGroup property
tomcontileslie f8a4366
SSS Decomp bugfix
tomcontileslie 2c1d259
SSS Decomp: add test
tomcontileslie 21e5eae
SSS Decomp: add 2 more tests
tomcontileslie 825aee3
SSS Decomp: address Wilf's comments
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Original file line number | Diff line number | Diff line change | ||||||||||||||||||||||
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@@ -327,3 +327,60 @@ function(S) | |||||||||||||||||||||||
UseIsomorphismRelation(H, G); | ||||||||||||||||||||||||
return H; | ||||||||||||||||||||||||
end); | ||||||||||||||||||||||||
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InstallMethod(IsomorphismSemigroup, | ||||||||||||||||||||||||
"for IsStrongSemilatticeOfSemigroups and a Clifford semigroup", | ||||||||||||||||||||||||
[IsStrongSemilatticeOfSemigroups, IsSemigroup and IsFinite], | ||||||||||||||||||||||||
function(filt, S) | ||||||||||||||||||||||||
local A, idemps, n, D, N, L, classes, idemp, DC, H, map, SSS, i, j; | ||||||||||||||||||||||||
# decomposes a finite Clifford semigroup S into a strong semilattice of | ||||||||||||||||||||||||
# groups and returns an SSS object. | ||||||||||||||||||||||||
if not (IsCliffordSemigroup(S) and IsFinite(S)) then | ||||||||||||||||||||||||
TryNextMethod(); | ||||||||||||||||||||||||
fi; | ||||||||||||||||||||||||
# There should be one idempotent per D-class, i.e. per semilattice element | ||||||||||||||||||||||||
# since the semilattice decomposition is by J-classes, and J = D here | ||||||||||||||||||||||||
A := Semigroup(Idempotents(S)); | ||||||||||||||||||||||||
idemps := Elements(A); | ||||||||||||||||||||||||
n := Size(idemps); | ||||||||||||||||||||||||
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# create semilattice | ||||||||||||||||||||||||
D := DigraphReflexiveTransitiveReduction(Digraph(NaturalPartialOrder(A))); | ||||||||||||||||||||||||
# currently wrong way round | ||||||||||||||||||||||||
D := DigraphReverse(D); | ||||||||||||||||||||||||
N := OutNeighbours(D); | ||||||||||||||||||||||||
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# populate list of semigroups in semilattice. | ||||||||||||||||||||||||
# keep a list of D-classes at the same time, to figure out where elements are | ||||||||||||||||||||||||
L := []; | ||||||||||||||||||||||||
classes := []; | ||||||||||||||||||||||||
for i in [1 .. n] do | ||||||||||||||||||||||||
idemp := idemps[i]; # the idempotent of this D-class | ||||||||||||||||||||||||
DC := DClass(S, idemp); | ||||||||||||||||||||||||
Add(L, Semigroup(DC)); | ||||||||||||||||||||||||
Add(classes, DC); | ||||||||||||||||||||||||
od; | ||||||||||||||||||||||||
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# populate list of homomorphisms | ||||||||||||||||||||||||
H := []; | ||||||||||||||||||||||||
for i in [1 .. n] do | ||||||||||||||||||||||||
idemp := idemps[i]; | ||||||||||||||||||||||||
Add(H, []); | ||||||||||||||||||||||||
for j in N[i] do | ||||||||||||||||||||||||
map := function(elm) | ||||||||||||||||||||||||
return idemp * elm; | ||||||||||||||||||||||||
end; | ||||||||||||||||||||||||
Add(H[i], MappingByFunction(L[j], L[i], map)); | ||||||||||||||||||||||||
od; | ||||||||||||||||||||||||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. It seems like there is no need to keep re-creating the function
Suggested change
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od; | ||||||||||||||||||||||||
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SSS := StrongSemilatticeOfSemigroups(D, L, H); | ||||||||||||||||||||||||
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return MagmaIsomorphismByFunctionsNC(S, | ||||||||||||||||||||||||
SSS, | ||||||||||||||||||||||||
x -> SSSE(SSS, | ||||||||||||||||||||||||
Position(classes, | ||||||||||||||||||||||||
DClass(S, x)), | ||||||||||||||||||||||||
x), | ||||||||||||||||||||||||
x -> x![3]); | ||||||||||||||||||||||||
end); |
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Original file line number | Diff line number | Diff line change | ||||
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@@ -102,3 +102,6 @@ DeclareProperty("IsSurjectiveSemigroup", IsSemigroup); | |||||
InstallTrueMethod(IsSurjectiveSemigroup, IsRegularSemigroup); | ||||||
InstallTrueMethod(IsSurjectiveSemigroup, IsMonoidAsSemigroup); | ||||||
InstallTrueMethod(IsSurjectiveSemigroup, IsIdempotentGenerated); | ||||||
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DeclareProperty("IsOrthogroup", IsSemigroup); | ||||||
DeclareSynonym("IsOrthoGroup", IsOrthogroup); | ||||||
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Suggested change
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You can directly get this via
(although I don't think that necessarily (or at all?) contains
x
inN[x]
for eachx
- if those are actually necessary, you will still want to add them or modify your code below to acts as if they are there).There was a problem hiding this comment.
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Thanks @wilfwilson, unfortunately I get a "no method found" error when running ReverseNaturalPartialOrder on some inputs - for example:
Although you are correct that we don't need
x
inN[x]
since the homomorphisms in this case are the identity, and the strong semilattice constructor is clever enough to fill that in upon creation.There was a problem hiding this comment.
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Also, I'm realising that taking the reflexive transitive reduction of
D
is redundant, since the SSS constructor effectively reverts this, so I'll remove that part