Combinatorial results for semigroups of order-preserving full transformations

A. Laradji; A. Umar

doi:10.1007/s00233-005-0553-6

Combinatorial results for semigroups of order-preserving full transformations

A. Laradji^*
, A. Umar

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

41 Scopus citations

Abstract

Let O_n be the semigroup of all order-preserving full transformations of a finite chain, say X_n = {1, 2, ..., n}, and for a given full transformation α: X_n → X_n let f(α) = |{x X_n: xα = x}|. In this note we obtain and discuss formulae for f(n,r,k) = |{α → O_n: f(α) = r ∧ max(Im α) = k}| and J(n,r,k) = |{α → O_n: |Im α| = r ∧ max(Im α) = k}|. We also obtain similar results for E(O_n), the set of idempotents of O_n.

Original language	English
Pages (from-to)	51-62
Number of pages	12
Journal	Semigroup Forum
Volume	72
Issue number	1
DOIs	https://doi.org/10.1007/s00233-005-0553-6
State	Published - Feb 2006

Keywords

Catalan number
Fibonacci number
Full transformation
Idempotent
Nilpotent
Order-decreasing
Order-preserving
Partial transformation
Semigroup

ASJC Scopus subject areas

Algebra and Number Theory

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10.1007/s00233-005-0553-6

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title = "Combinatorial results for semigroups of order-preserving full transformations",

abstract = "Let On be the semigroup of all order-preserving full transformations of a finite chain, say Xn = \{1, 2, ..., n\}, and for a given full transformation α: Xn → Xn let f(α) = |\{x Xn: xα = x\}|. In this note we obtain and discuss formulae for f(n,r,k) = |\{α → On: f(α) = r ∧ max(Im α) = k\}| and J(n,r,k) = |\{α → On: |Im α| = r ∧ max(Im α) = k\}|. We also obtain similar results for E(On), the set of idempotents of On.",

keywords = "Catalan number, Fibonacci number, Full transformation, Idempotent, Nilpotent, Order-decreasing, Order-preserving, Partial transformation, Semigroup",

author = "A. Laradji and A. Umar",

year = "2006",

month = feb,

doi = "10.1007/s00233-005-0553-6",

language = "English",

volume = "72",

pages = "51--62",

journal = "Semigroup Forum",

issn = "0037-1912",

number = "1",

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TY - JOUR

T1 - Combinatorial results for semigroups of order-preserving full transformations

AU - Laradji, A.

AU - Umar, A.

PY - 2006/2

Y1 - 2006/2

N2 - Let On be the semigroup of all order-preserving full transformations of a finite chain, say Xn = {1, 2, ..., n}, and for a given full transformation α: Xn → Xn let f(α) = |{x Xn: xα = x}|. In this note we obtain and discuss formulae for f(n,r,k) = |{α → On: f(α) = r ∧ max(Im α) = k}| and J(n,r,k) = |{α → On: |Im α| = r ∧ max(Im α) = k}|. We also obtain similar results for E(On), the set of idempotents of On.

AB - Let On be the semigroup of all order-preserving full transformations of a finite chain, say Xn = {1, 2, ..., n}, and for a given full transformation α: Xn → Xn let f(α) = |{x Xn: xα = x}|. In this note we obtain and discuss formulae for f(n,r,k) = |{α → On: f(α) = r ∧ max(Im α) = k}| and J(n,r,k) = |{α → On: |Im α| = r ∧ max(Im α) = k}|. We also obtain similar results for E(On), the set of idempotents of On.

KW - Catalan number

KW - Fibonacci number

KW - Full transformation

KW - Idempotent

KW - Nilpotent

KW - Order-decreasing

KW - Order-preserving

KW - Partial transformation

KW - Semigroup

UR - https://www.scopus.com/pages/publications/33645529135

U2 - 10.1007/s00233-005-0553-6

DO - 10.1007/s00233-005-0553-6

M3 - Article

AN - SCOPUS:33645529135

SN - 0037-1912

VL - 72

SP - 51

EP - 62

JO - Semigroup Forum

JF - Semigroup Forum

IS - 1

ER -

Combinatorial results for semigroups of order-preserving full transformations

Abstract

Keywords

ASJC Scopus subject areas

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