new maximal subsemigroups of the semigroup of all ... · pdf filenew maximal subsemigroups of...
TRANSCRIPT
![Page 1: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/1.jpg)
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
![Page 2: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/2.jpg)
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.
Jörg Koppitz (Institute) maximal subsemigroups 03/06 2 / 11
![Page 3: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/3.jpg)
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.
Jörg Koppitz (Institute) maximal subsemigroups 03/06 2 / 11
![Page 4: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/4.jpg)
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.
Jörg Koppitz (Institute) maximal subsemigroups 03/06 2 / 11
![Page 5: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/5.jpg)
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.
Jörg Koppitz (Institute) maximal subsemigroups 03/06 2 / 11
![Page 6: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/6.jpg)
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.
Jörg Koppitz (Institute) maximal subsemigroups 03/06 3 / 11
![Page 7: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/7.jpg)
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 Ω
the stabilizer of a nite partition of Ωthe stabilizer of an ultralter on Ω.
Jörg Koppitz (Institute) maximal subsemigroups 03/06 3 / 11
![Page 8: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/8.jpg)
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 Ω
the stabilizer of an ultralter on Ω.
Jörg Koppitz (Institute) maximal subsemigroups 03/06 3 / 11
![Page 9: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/9.jpg)
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.
Jörg Koppitz (Institute) maximal subsemigroups 03/06 3 / 11
![Page 10: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/10.jpg)
...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
Jörg Koppitz (Institute) maximal subsemigroups 03/06 4 / 11
![Page 11: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/11.jpg)
...containing a particular semigroup U
Let Ω be countable
ΩΩ semigroup of all mappings on the set Ω
U ΩΩ
W ΩΩ, where each α 2 U is a generator modulo WW S ΩΩ
ProblemCharacterization of all maximal subsemigroups of S
Jörg Koppitz (Institute) maximal subsemigroups 03/06 4 / 11
![Page 12: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/12.jpg)
...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
Jörg Koppitz (Institute) maximal subsemigroups 03/06 4 / 11
![Page 13: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/13.jpg)
...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 W
W S ΩΩ
ProblemCharacterization of all maximal subsemigroups of S
Jörg Koppitz (Institute) maximal subsemigroups 03/06 4 / 11
![Page 14: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/14.jpg)
...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
Jörg Koppitz (Institute) maximal subsemigroups 03/06 4 / 11
![Page 15: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/15.jpg)
...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
Jörg Koppitz (Institute) maximal subsemigroups 03/06 4 / 11
![Page 16: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/16.jpg)
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
Jörg Koppitz (Institute) maximal subsemigroups 03/06 5 / 11
![Page 17: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/17.jpg)
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
Jörg Koppitz (Institute) maximal subsemigroups 03/06 5 / 11
![Page 18: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/18.jpg)
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
Jörg Koppitz (Institute) maximal subsemigroups 03/06 5 / 11
![Page 19: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/19.jpg)
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.
Jörg Koppitz (Institute) maximal subsemigroups 03/06 6 / 11
![Page 20: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/20.jpg)
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
Jörg Koppitz (Institute) maximal subsemigroups 03/06 7 / 11
![Page 21: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/21.jpg)
The maximal subsemigroups containing the symmetricgroup
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
= @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
Jörg Koppitz (Institute) maximal subsemigroups 03/06 7 / 11
![Page 22: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/22.jpg)
The maximal subsemigroups containing the symmetricgroup
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)
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
Jörg Koppitz (Institute) maximal subsemigroups 03/06 7 / 11
![Page 23: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/23.jpg)
The maximal subsemigroups containing the symmetricgroup
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)
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
Jörg Koppitz (Institute) maximal subsemigroups 03/06 7 / 11
![Page 24: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/24.jpg)
The maximal subsemigroups containing the symmetricgroup
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)
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
Jörg Koppitz (Institute) maximal subsemigroups 03/06 7 / 11
![Page 25: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/25.jpg)
The maximal subsemigroups containing the symmetricgroup
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)
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
Jörg Koppitz (Institute) maximal subsemigroups 03/06 7 / 11
![Page 26: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/26.jpg)
The maximal subsemigroups containing the symmetricgroup
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)
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
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
Jörg Koppitz (Institute) maximal subsemigroups 03/06 7 / 11
![Page 27: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/27.jpg)
The maximal subsemigroups containing the symmetricgroup
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)
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
Jörg Koppitz (Institute) maximal subsemigroups 03/06 7 / 11
![Page 28: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/28.jpg)
The maximal subsemigroups containing the symmetricgroup
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)
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
Jörg Koppitz (Institute) maximal subsemigroups 03/06 7 / 11
![Page 29: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/29.jpg)
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).
Jörg Koppitz (Institute) maximal subsemigroups 03/06 8 / 11
![Page 30: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/30.jpg)
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).
Jörg Koppitz (Institute) maximal subsemigroups 03/06 8 / 11
![Page 31: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/31.jpg)
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).
Jörg Koppitz (Institute) maximal subsemigroups 03/06 8 / 11
![Page 32: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/32.jpg)
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)).
Jörg Koppitz (Institute) maximal subsemigroups 03/06 9 / 11
![Page 33: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/33.jpg)
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)).
Jörg Koppitz (Institute) maximal subsemigroups 03/06 9 / 11
![Page 34: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/34.jpg)
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)).
Jörg Koppitz (Institute) maximal subsemigroups 03/06 9 / 11
![Page 35: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/35.jpg)
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)).
Jörg Koppitz (Institute) maximal subsemigroups 03/06 9 / 11
![Page 36: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/36.jpg)
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)).
Jörg Koppitz (Institute) maximal subsemigroups 03/06 10 / 11
![Page 37: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/37.jpg)
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)).
Jörg Koppitz (Institute) maximal subsemigroups 03/06 10 / 11
![Page 38: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/38.jpg)
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)).
Jörg Koppitz (Institute) maximal subsemigroups 03/06 10 / 11
![Page 39: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/39.jpg)
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)).
Jörg Koppitz (Institute) maximal subsemigroups 03/06 10 / 11
![Page 40: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/40.jpg)
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)).
Jörg Koppitz (Institute) maximal subsemigroups 03/06 11 / 11
![Page 41: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/41.jpg)
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)).
Jörg Koppitz (Institute) maximal subsemigroups 03/06 11 / 11
![Page 42: 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](https://reader031.vdocuments.net/reader031/viewer/2022030409/5a94cd587f8b9ab6188c004e/html5/thumbnails/42.jpg)
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)).
Jörg Koppitz (Institute) maximal subsemigroups 03/06 11 / 11