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The rank of the semigroup of all order-preservingtransformations on a �nite fence
Jörg Koppitz (joit project with Vitor Fernandes and TiwadeeMusunthia)
Institute of Mathematics and Informatics,Bulgarian Academy of Sciences, So�a, Bulgaria
October 12 2017
Jörg Koppitz (joit project with Vitor Fernandes and Tiwadee Musunthia) (Institute)The rank of the semigroup of all order-preserving transformations on a �nite fenceOctober 12 2017 1 / 12
Order-preserving Transformations
n 2 N
Tn - monoid of all full transformations on n := f1, . . . , ng� partial order on n
α 2 Tn is order-preserving if
x � y ) x � y
important example: � is a chain
On - set of all order-preserving transformations on a chainn = f1 < � � � < ng
Jörg Koppitz (joit project with Vitor Fernandes and Tiwadee Musunthia) (Institute)The rank of the semigroup of all order-preserving transformations on a �nite fenceOctober 12 2017 2 / 12
History
1962 Aizen�tat: On has only one non-trivial automorphism
1971 Howie cardinal and number of idempotents for On1992 Howie & Gomes rank and idempotent rank for On
Jörg Koppitz (joit project with Vitor Fernandes and Tiwadee Musunthia) (Institute)The rank of the semigroup of all order-preserving transformations on a �nite fenceOctober 12 2017 3 / 12
History
1962 Aizen�tat: On has only one non-trivial automorphism1971 Howie cardinal and number of idempotents for On
1992 Howie & Gomes rank and idempotent rank for On
Jörg Koppitz (joit project with Vitor Fernandes and Tiwadee Musunthia) (Institute)The rank of the semigroup of all order-preserving transformations on a �nite fenceOctober 12 2017 3 / 12
History
1962 Aizen�tat: On has only one non-trivial automorphism1971 Howie cardinal and number of idempotents for On1992 Howie & Gomes rank and idempotent rank for On
Jörg Koppitz (joit project with Vitor Fernandes and Tiwadee Musunthia) (Institute)The rank of the semigroup of all order-preserving transformations on a �nite fenceOctober 12 2017 3 / 12
Fences
.
A non-linear order close to a linear order: zig-zag order (fence)
A pair (n,�) is called zig-zag poset or fence if
1 � 2 � 3 � � � ��� n� 1 � n if n is odd� n� 1 � n if n is even
or dually
every element in a fence is either minimal or maximal
Jörg Koppitz (joit project with Vitor Fernandes and Tiwadee Musunthia) (Institute)The rank of the semigroup of all order-preserving transformations on a �nite fenceOctober 12 2017 4 / 12
Fences
.
A non-linear order close to a linear order: zig-zag order (fence)
A pair (n,�) is called zig-zag poset or fence if
1 � 2 � 3 � � � ��� n� 1 � n if n is odd� n� 1 � n if n is even
or dually
every element in a fence is either minimal or maximal
Jörg Koppitz (joit project with Vitor Fernandes and Tiwadee Musunthia) (Institute)The rank of the semigroup of all order-preserving transformations on a �nite fenceOctober 12 2017 4 / 12
Fences
.
A non-linear order close to a linear order: zig-zag order (fence)
A pair (n,�) is called zig-zag poset or fence if
1 � 2 � 3 � � � ��� n� 1 � n if n is odd� n� 1 � n if n is even
or dually
every element in a fence is either minimal or maximal
Jörg Koppitz (joit project with Vitor Fernandes and Tiwadee Musunthia) (Institute)The rank of the semigroup of all order-preserving transformations on a �nite fenceOctober 12 2017 4 / 12
About Fence-preserving Transformations
1991/92 Currie & Visentin and Rutkowski,respectively �rst study
Currie & Visentin number of order-preserving transformations of afence (n is even) 1991
for any n by Rutkowski 1992
S. Srithus et al Regularity 2015
Dimitrova et al. injective order-preserving partial transformations on afence 2017
Laddawan et al. Regular subsemigroups of order-preservingtransformations on an in�nite fence 2017
Jörg Koppitz (joit project with Vitor Fernandes and Tiwadee Musunthia) (Institute)The rank of the semigroup of all order-preserving transformations on a �nite fenceOctober 12 2017 5 / 12
About Fence-preserving Transformations
1991/92 Currie & Visentin and Rutkowski,respectively �rst study
Currie & Visentin number of order-preserving transformations of afence (n is even) 1991
for any n by Rutkowski 1992
S. Srithus et al Regularity 2015
Dimitrova et al. injective order-preserving partial transformations on afence 2017
Laddawan et al. Regular subsemigroups of order-preservingtransformations on an in�nite fence 2017
Jörg Koppitz (joit project with Vitor Fernandes and Tiwadee Musunthia) (Institute)The rank of the semigroup of all order-preserving transformations on a �nite fenceOctober 12 2017 5 / 12
About Fence-preserving Transformations
1991/92 Currie & Visentin and Rutkowski,respectively �rst study
Currie & Visentin number of order-preserving transformations of afence (n is even) 1991
for any n by Rutkowski 1992
S. Srithus et al Regularity 2015
Dimitrova et al. injective order-preserving partial transformations on afence 2017
Laddawan et al. Regular subsemigroups of order-preservingtransformations on an in�nite fence 2017
Jörg Koppitz (joit project with Vitor Fernandes and Tiwadee Musunthia) (Institute)The rank of the semigroup of all order-preserving transformations on a �nite fenceOctober 12 2017 5 / 12
About Fence-preserving Transformations
1991/92 Currie & Visentin and Rutkowski,respectively �rst study
Currie & Visentin number of order-preserving transformations of afence (n is even) 1991
for any n by Rutkowski 1992
S. Srithus et al Regularity 2015
Dimitrova et al. injective order-preserving partial transformations on afence 2017
Laddawan et al. Regular subsemigroups of order-preservingtransformations on an in�nite fence 2017
Jörg Koppitz (joit project with Vitor Fernandes and Tiwadee Musunthia) (Institute)The rank of the semigroup of all order-preserving transformations on a �nite fenceOctober 12 2017 5 / 12
About Fence-preserving Transformations
1991/92 Currie & Visentin and Rutkowski,respectively �rst study
Currie & Visentin number of order-preserving transformations of afence (n is even) 1991
for any n by Rutkowski 1992
S. Srithus et al Regularity 2015
Dimitrova et al. injective order-preserving partial transformations on afence 2017
Laddawan et al. Regular subsemigroups of order-preservingtransformations on an in�nite fence 2017
Jörg Koppitz (joit project with Vitor Fernandes and Tiwadee Musunthia) (Institute)The rank of the semigroup of all order-preserving transformations on a �nite fenceOctober 12 2017 5 / 12
About Fence-preserving Transformations
1991/92 Currie & Visentin and Rutkowski,respectively �rst study
Currie & Visentin number of order-preserving transformations of afence (n is even) 1991
for any n by Rutkowski 1992
S. Srithus et al Regularity 2015
Dimitrova et al. injective order-preserving partial transformations on afence 2017
Laddawan et al. Regular subsemigroups of order-preservingtransformations on an in�nite fence 2017
Jörg Koppitz (joit project with Vitor Fernandes and Tiwadee Musunthia) (Institute)The rank of the semigroup of all order-preserving transformations on a �nite fenceOctober 12 2017 5 / 12
The Monoid
T F n - submonoid of Tn of all order-preserving transformations of thefence (n,�)
w.l.o.g. let 1 � 2 � 3 � � � �idn - identity mapping on n
If n is even then idn is the only permutation in T F nIf n is odd then idn and
γn :=�1 2 � � � nn n� 1 � � � 1
�are the permutations on T F n.All constant mappings on n are in T F n.
Jörg Koppitz (joit project with Vitor Fernandes and Tiwadee Musunthia) (Institute)The rank of the semigroup of all order-preserving transformations on a �nite fenceOctober 12 2017 6 / 12
The Monoid
T F n - submonoid of Tn of all order-preserving transformations of thefence (n,�)w.l.o.g. let 1 � 2 � 3 � � � �
idn - identity mapping on n
If n is even then idn is the only permutation in T F nIf n is odd then idn and
γn :=�1 2 � � � nn n� 1 � � � 1
�are the permutations on T F n.All constant mappings on n are in T F n.
Jörg Koppitz (joit project with Vitor Fernandes and Tiwadee Musunthia) (Institute)The rank of the semigroup of all order-preserving transformations on a �nite fenceOctober 12 2017 6 / 12
The Monoid
T F n - submonoid of Tn of all order-preserving transformations of thefence (n,�)w.l.o.g. let 1 � 2 � 3 � � � �idn - identity mapping on n
If n is even then idn is the only permutation in T F nIf n is odd then idn and
γn :=�1 2 � � � nn n� 1 � � � 1
�are the permutations on T F n.All constant mappings on n are in T F n.
Jörg Koppitz (joit project with Vitor Fernandes and Tiwadee Musunthia) (Institute)The rank of the semigroup of all order-preserving transformations on a �nite fenceOctober 12 2017 6 / 12
The Monoid
T F n - submonoid of Tn of all order-preserving transformations of thefence (n,�)w.l.o.g. let 1 � 2 � 3 � � � �idn - identity mapping on n
If n is even then idn is the only permutation in T F n
If n is odd then idn and
γn :=�1 2 � � � nn n� 1 � � � 1
�are the permutations on T F n.All constant mappings on n are in T F n.
Jörg Koppitz (joit project with Vitor Fernandes and Tiwadee Musunthia) (Institute)The rank of the semigroup of all order-preserving transformations on a �nite fenceOctober 12 2017 6 / 12
The Monoid
T F n - submonoid of Tn of all order-preserving transformations of thefence (n,�)w.l.o.g. let 1 � 2 � 3 � � � �idn - identity mapping on n
If n is even then idn is the only permutation in T F nIf n is odd then idn and
γn :=�1 2 � � � nn n� 1 � � � 1
�are the permutations on T F n.
All constant mappings on n are in T F n.
Jörg Koppitz (joit project with Vitor Fernandes and Tiwadee Musunthia) (Institute)The rank of the semigroup of all order-preserving transformations on a �nite fenceOctober 12 2017 6 / 12
The Monoid
T F n - submonoid of Tn of all order-preserving transformations of thefence (n,�)w.l.o.g. let 1 � 2 � 3 � � � �idn - identity mapping on n
If n is even then idn is the only permutation in T F nIf n is odd then idn and
γn :=�1 2 � � � nn n� 1 � � � 1
�are the permutations on T F n.All constant mappings on n are in T F n.
Jörg Koppitz (joit project with Vitor Fernandes and Tiwadee Musunthia) (Institute)The rank of the semigroup of all order-preserving transformations on a �nite fenceOctober 12 2017 6 / 12
Characterization of the Order-Preserving Transformations
Theorem (F&K&M)Let α 2 Tn. Then α 2 T F n if and only if(i) jxα� (x + 1)αj � 1 for all x 2 f1, . . . , n� 1g.(ii) x and xα have the same parity or (x � 1)α = xα = (x + 1)α for allx 2 f2, . . . , n� 1g.
Corollary
If α 2 Tn then α 2 T F n if and only if Im α = fk, k + 1, . . . , lg for some1 � k � l � n. (Im α is the image of α)
Jörg Koppitz (joit project with Vitor Fernandes and Tiwadee Musunthia) (Institute)The rank of the semigroup of all order-preserving transformations on a �nite fenceOctober 12 2017 7 / 12
Idempotents
A transformation α is called idempotent if αα = α.
α 2 Tn is idempotent if and only if Im α = fx 2 n : xα = xg.En - set of all idempotents in T F n.
E1 = T F 1 =��
11
��E2 = T F 2 =
��1 21 2
�,
�121
�,
�122
��= T2 n
��1 22 1
��
Jörg Koppitz (joit project with Vitor Fernandes and Tiwadee Musunthia) (Institute)The rank of the semigroup of all order-preserving transformations on a �nite fenceOctober 12 2017 8 / 12
Idempotents
A transformation α is called idempotent if αα = α.
α 2 Tn is idempotent if and only if Im α = fx 2 n : xα = xg.
En - set of all idempotents in T F n.
E1 = T F 1 =��
11
��E2 = T F 2 =
��1 21 2
�,
�121
�,
�122
��= T2 n
��1 22 1
��
Jörg Koppitz (joit project with Vitor Fernandes and Tiwadee Musunthia) (Institute)The rank of the semigroup of all order-preserving transformations on a �nite fenceOctober 12 2017 8 / 12
Idempotents
A transformation α is called idempotent if αα = α.
α 2 Tn is idempotent if and only if Im α = fx 2 n : xα = xg.En - set of all idempotents in T F n.
E1 = T F 1 =��
11
��E2 = T F 2 =
��1 21 2
�,
�121
�,
�122
��= T2 n
��1 22 1
��
Jörg Koppitz (joit project with Vitor Fernandes and Tiwadee Musunthia) (Institute)The rank of the semigroup of all order-preserving transformations on a �nite fenceOctober 12 2017 8 / 12
Idempotents
A transformation α is called idempotent if αα = α.
α 2 Tn is idempotent if and only if Im α = fx 2 n : xα = xg.En - set of all idempotents in T F n.
E1 = T F 1 =��
11
��
E2 = T F 2 =��
1 21 2
�,
�121
�,
�122
��= T2 n
��1 22 1
��
Jörg Koppitz (joit project with Vitor Fernandes and Tiwadee Musunthia) (Institute)The rank of the semigroup of all order-preserving transformations on a �nite fenceOctober 12 2017 8 / 12
Idempotents
A transformation α is called idempotent if αα = α.
α 2 Tn is idempotent if and only if Im α = fx 2 n : xα = xg.En - set of all idempotents in T F n.
E1 = T F 1 =��
11
��E2 = T F 2 =
��1 21 2
�,
�121
�,
�122
��= T2 n
��1 22 1
��
Jörg Koppitz (joit project with Vitor Fernandes and Tiwadee Musunthia) (Institute)The rank of the semigroup of all order-preserving transformations on a �nite fenceOctober 12 2017 8 / 12
Number of Idempotents
Theorem (F&K&M)
jEn j =n
∑k=1
n�k∑p=0A(k, p)A((n+ 1)� (k + p), p), where
A(m, p) =m�1∑r=0
jP(p, r)j jK (m, r)j.
Jörg Koppitz (joit project with Vitor Fernandes and Tiwadee Musunthia) (Institute)The rank of the semigroup of all order-preserving transformations on a �nite fenceOctober 12 2017 9 / 12
Results by GAP and Formula
n jEn j jT F n j1 1 12 3 33 8 114 19 315 44 996 98 2757 218 8118 474 2199
Jörg Koppitz (joit project with Vitor Fernandes and Tiwadee Musunthia) (Institute)The rank of the semigroup of all order-preserving transformations on a �nite fenceOctober 12 2017 10 / 12
Rank of Semigroup
S � Tn
rankS := minfjAj : A � S , hAi = Sg (rank of S)
Theorem (F&K&M)
rank(T F n) =
8>>><>>>:32 (n� 1) +
n�5∑k=2
�� n�1�2k3
�� 1
�if n � 3 is odd
3n� 8+n�7∑k=2
�� n�1�k3
�� 1
�if n is even.
FactWe can provide a minimal size generating set for T F n, n is even n is odd.
Jörg Koppitz (joit project with Vitor Fernandes and Tiwadee Musunthia) (Institute)The rank of the semigroup of all order-preserving transformations on a �nite fenceOctober 12 2017 11 / 12
Rank of Semigroup
S � TnrankS := minfjAj : A � S , hAi = Sg (rank of S)
Theorem (F&K&M)
rank(T F n) =
8>>><>>>:32 (n� 1) +
n�5∑k=2
�� n�1�2k3
�� 1
�if n � 3 is odd
3n� 8+n�7∑k=2
�� n�1�k3
�� 1
�if n is even.
FactWe can provide a minimal size generating set for T F n, n is even n is odd.
Jörg Koppitz (joit project with Vitor Fernandes and Tiwadee Musunthia) (Institute)The rank of the semigroup of all order-preserving transformations on a �nite fenceOctober 12 2017 11 / 12
Rank of Semigroup
S � TnrankS := minfjAj : A � S , hAi = Sg (rank of S)
Theorem (F&K&M)
rank(T F n) =
8>>><>>>:32 (n� 1) +
n�5∑k=2
�� n�1�2k3
�� 1
�if n � 3 is odd
3n� 8+n�7∑k=2
�� n�1�k3
�� 1
�if n is even.
FactWe can provide a minimal size generating set for T F n, n is even n is odd.
Jörg Koppitz (joit project with Vitor Fernandes and Tiwadee Musunthia) (Institute)The rank of the semigroup of all order-preserving transformations on a �nite fenceOctober 12 2017 11 / 12
Rank of Semigroup
S � TnrankS := minfjAj : A � S , hAi = Sg (rank of S)
Theorem (F&K&M)
rank(T F n) =
8>>><>>>:32 (n� 1) +
n�5∑k=2
�� n�1�2k3
�� 1
�if n � 3 is odd
3n� 8+n�7∑k=2
�� n�1�k3
�� 1
�if n is even.
FactWe can provide a minimal size generating set for T F n, n is even n is odd.
Jörg Koppitz (joit project with Vitor Fernandes and Tiwadee Musunthia) (Institute)The rank of the semigroup of all order-preserving transformations on a �nite fenceOctober 12 2017 11 / 12