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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


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


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


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


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


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


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.


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


γn :=�1 2 � � � nn n� 1 � � � 1



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


γn :=�1 2 � � � nn n� 1 � � � 1



The Monoid


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



The Monoid



γn :=�1 2 � � � nn n� 1 � � � 1

�are the permutations on T F n.

All constant mappings on n are in T F n.


The Monoid



γn :=�1 2 � � � nn n� 1 � � � 1



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)


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

��


Idempotents


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

��


Idempotents



E1 = T F 1 =��

11

��E2 = T F 2 =

��1 21 2

�,

�121

�,

�122

��= T2 n

��1 22 1

��


Idempotents



E1 = T F 1 =��

11

��

E2 = T F 2 =��

1 21 2

�,

�121

�,

�122

��= T2 n

��1 22 1

��


Idempotents



E1 = T F 1 =��

11

��E2 = T F 2 =

��1 21 2

�,

�121

�,

�122

��= T2 n

��1 22 1

��


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.


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


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.


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.


Thank you very much
