New Maximal Subsemigroups of the Semigroup of all ... · PDF fileNew Maximal Subsemigroups of...

New Maximal Subsemigroups of the Semigroup of allTransformations on a countable set

Jörg Koppitz

Potsdam University

March 2012

Jörg Koppitz (Institute) maximal subsemigroups 03/06 1 / 11

History

2002 L. Heindorf : The maximal clones on countable sets that includeall permutations

2005 M. Pinsker: Clones on uncountable sets that include allpermutations

April 2011 J. East, D. Mitchell, Y. Péresse: Maximal subsemigroupsof the semigroup of all mappings on an innite set containing allpermutations

September 2011 J. East, D. Mitchell, Y. Péresse: Maximalsubsemigroups of the semigroup of all mappings on an innite set.

History

East, Mitchell, Péresse

Classication of the maximal subsemigroups of the semigroup of allmappings on an innite set Ω that contains one of the followingsubgroups of the symmetric group on Ω:

setwise stabilizer of a non-empty nite subset of Ωthe stabilizer of a nite partition of Ωthe stabilizer of an ultralter on Ω.

Classication of the maximal subsemigroups of the semigroup of allmappings on an innite set Ω that contains one of the followingsubgroups of the symmetric group on Ω:setwise stabilizer of a non-empty nite subset of Ω

the stabilizer of a nite partition of Ωthe stabilizer of an ultralter on Ω.

Classication of the maximal subsemigroups of the semigroup of allmappings on an innite set Ω that contains one of the followingsubgroups of the symmetric group on Ω:setwise stabilizer of a non-empty nite subset of Ωthe stabilizer of a nite partition of Ω

the stabilizer of an ultralter on Ω.

Classication of the maximal subsemigroups of the semigroup of allmappings on an innite set Ω that contains one of the followingsubgroups of the symmetric group on Ω:setwise stabilizer of a non-empty nite subset of Ωthe stabilizer of a nite partition of Ωthe stabilizer of an ultralter on Ω.

...containing a particular semigroup U

Let Ω be countable

ΩΩ semigroup of all mappings on the set ΩU ΩΩ

W ΩΩ, where each α 2 U is a generator modulo WW S ΩΩ

ProblemCharacterization of all maximal subsemigroups of S

Let Ω be countable

ΩΩ semigroup of all mappings on the set Ω

U ΩΩ

Let Ω be countable

W ΩΩ, where each α 2 U is a generator modulo W

W S ΩΩ

Let Ω be countable

Notations

we dene a set H(U,W )

Denition

For M P(ΩΩ), let J(M) be the set of all A [M with

8m 2 M(A\m 6= ∅)&8a 2 A9m 2 M(A\m = fag)

Denition

For U ΩΩ and W ΩΩ, we putGen(U) := fA ΩΩ j A is nite and hAi \U 6= ∅g andH(U,W ) := fA ΩΩ nW j A 2 J(Gen(U))g

Notations

Denition

8m 2 M(A\m 6= ∅)&8a 2 A9m 2 M(A\m = fag)

Denition

Notations

Denition

8m 2 M(A\m 6= ∅)&8a 2 A9m 2 M(A\m = fag)

Denition

Main theorem

Theorem

Let W S ΩΩ and U ΩΩ such that each α 2 U is a generatormodulo W . Then the following statements are equivalent:(i) S is maximal.(ii) There is a set H 2 H(U,W ) with S = ΩΩ nH.

The maximal subsemigroups containing the symmetricgroup

Inj(Ω) the set of all injective but not surjective mappings on Ω

Sur(Ω) the set of all surjective but not injective mappings on ΩCp(Ω) := fα 2 ΩΩ j rank(α) = @0 and k(α) = @0gk(α) :=

fx 2 imα jxα1

= @0 (innite contraction index of α)d(α) := jΩ n imαj (defect of α)c(α) := ∑

x2imα(xα1

1) (collapse of α)

IF (Ω) := fα 2 ΩΩ j rank(α) = @0, c(α) = @0, and d(α) < @0gFI (Ω) := fα 2 ΩΩ j rank(α) = @0, d(α) = @0, and c(α) < @0g

Theorem(L. Heindorf 2002)Let S ΩΩ containing the symmetric group. S is maximal i¤S = ΩΩ nH for some H 2 fInj(Ω),Sur(Ω),Cp(Ω), IF (Ω),FI (Ω)g

Inj(Ω) the set of all injective but not surjective mappings on ΩSur(Ω) the set of all surjective but not injective mappings on Ω

Cp(Ω) := fα 2 ΩΩ j rank(α) = @0 and k(α) = @0gk(α) :=

fx 2 imα jxα1

x2imα(xα1

1) (collapse of α)

Inj(Ω) the set of all injective but not surjective mappings on ΩSur(Ω) the set of all surjective but not injective mappings on ΩCp(Ω) := fα 2 ΩΩ j rank(α) = @0 and k(α) = @0g

k(α) :=fx 2 imα j

xα1 = @0 (innite contraction index of α)

d(α) := jΩ n imαj (defect of α)c(α) := ∑

x2imα(xα1

1) (collapse of α)

Inj(Ω) the set of all injective but not surjective mappings on ΩSur(Ω) the set of all surjective but not injective mappings on ΩCp(Ω) := fα 2 ΩΩ j rank(α) = @0 and k(α) = @0gk(α) :=

fx 2 imα jxα1

= @0 (innite contraction index of α)

d(α) := jΩ n imαj (defect of α)c(α) := ∑

x2imα(xα1

1) (collapse of α)

fx 2 imα jxα1

= @0 (innite contraction index of α)d(α) := jΩ n imαj (defect of α)

c(α) := ∑x2imα

1) (collapse of α)

fx 2 imα jxα1

x2imα(xα1

1) (collapse of α)

fx 2 imα jxα1

x2imα(xα1

1) (collapse of α)

IF (Ω) := fα 2 ΩΩ j rank(α) = @0, c(α) = @0, and d(α) < @0g

FI (Ω) := fα 2 ΩΩ j rank(α) = @0, d(α) = @0, and c(α) < @0g

fx 2 imα jxα1

x2imα(xα1

1) (collapse of α)

fx 2 imα jxα1

x2imα(xα1

1) (collapse of α)

Sur(X) and Inj(X)

The maximal subsemigroups containing Inj(Ω) or Sur(Ω)

Theorem

Let S ΩΩ containing Sur(Ω). S is maximal i¤ S = ΩΩ n Inj(Ω) orS = ΩΩ n FI (Ω).

Theorem

Let S ΩΩ containing Inj(Ω). S is maximal i¤ S = ΩΩ n Sur(Ω) orS = ΩΩ n IF (Ω) or S = ΩΩ n Cp(Ω).

Sur(X) and Inj(X)

Theorem

Sur(X) and Inj(X)

Theorem

maximal subsemigroups containing FI (Ω) (using main theorem)

FI (Ω) is a subsemigroup of ΩΩ.

LemmaEach α 2 Cp(Ω) \ Sur(Ω) is a generator modulo FI (Ω).

Theorem

Let S ΩΩ containing FI (Ω). S is maximal i¤ S = ΩΩ nH for someH 2 H(Cp(Ω) \ Sur(Ω),FI (Ω)).

Theorem

maximal subsemigroups containing IF (Ω) (using main theorem)

IF (Ω) is a subsemigroup of ΩΩ.

LemmaEach α 2 FI (Ω) \ Inj(Ω) is a generator modulo IF (Ω).

Theorem

Let S ΩΩ containing IF (Ω). S is maximal i¤ S = ΩΩ nH for someH 2 H(Inj(Ω) \ FI (Ω), IF (Ω)).

Theorem

innite contraction index

maximal subsemigroups containing Cp(Ω) (using main theorem)

LemmaEach α 2 FI (Ω) \ Inj(Ω) is a generator modulo hCp(Ω)i.

Theorem

Let S ΩΩ containing IF (Ω). S is maximal i¤ S = ΩΩ nH for someH 2 H(Inj(Ω) \ FI (Ω),Cp(Ω)).

Theorem

New Maximal Subsemigroups of the Semigroup of all ... · PDF fileNew Maximal Subsemigroups of...

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