New Maximal Subsemigroups of the Semigroup of all ... · PDF fileNew Maximal Subsemigroups of the Semigroup of all Transformations on a countable set Jörg Koppitz Potsdam University

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






History






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






Let Ω be countable


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

W S ΩΩ




Let Ω be countable






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

(xα1

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


Theorem



Sur(X) and Inj(X)


Theorem


Theorem



FI(X)

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

Lemma

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)).


FI(X)


Lemma



Theorem



FI(X)


Lemma



Theorem



FI(X)


Lemma



Theorem



IF(X)

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

Lemma

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)).


IF(X)


Lemma



Theorem



IF(X)


Lemma



Theorem



IF(X)


Lemma



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






Theorem



Documents

New Maximal Subsemigroups of the Semigroup of all ... · PDF fileNew Maximal Subsemigroups of the Semigroup of all Transformations on a countable set Jörg Koppitz Potsdam University