Combinatorial results for semigroups of order-preserving full transformations

A. Laradji, A. Umar

Research output: Contribution to journalArticle

24 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)51-62
Number of pages12
JournalSemigroup Forum
Volume72
Issue number1
DOIs
Publication statusPublished - Feb 2006

Fingerprint

Semigroup
Idempotent

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Combinatorial results for semigroups of order-preserving full transformations. / Laradji, A.; Umar, A.

In: Semigroup Forum, Vol. 72, No. 1, 02.2006, p. 51-62.

Research output: Contribution to journalArticle

Laradji, A. ; Umar, A. / Combinatorial results for semigroups of order-preserving full transformations. In: Semigroup Forum. 2006 ; Vol. 72, No. 1. pp. 51-62.
@article{c1623db2e70143feb830cbac989202e1,
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 = "2",
doi = "10.1007/s00233-005-0553-6",
language = "English",
volume = "72",
pages = "51--62",
journal = "Semigroup Forum",
issn = "0037-1912",
publisher = "Springer New York",
number = "1",

}

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 - http://www.scopus.com/inward/record.url?scp=33645529135&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33645529135&partnerID=8YFLogxK

U2 - 10.1007/s00233-005-0553-6

DO - 10.1007/s00233-005-0553-6

M3 - Article

AN - SCOPUS:33645529135

VL - 72

SP - 51

EP - 62

JO - Semigroup Forum

JF - Semigroup Forum

SN - 0037-1912

IS - 1

ER -