On the ranks of certain semigroups of order-preserving transformations
type: Item
instance of: scholarly article
title: On the ranks of certain semigroups of order-preserving transformations
zbMATH Open document ID: 0769.20029
DOI: 10.1007/BF03025769
author: Gracinda M. S. Gomes
author: John M. Howie
published in: Semigroup Forum
publication date: 1993-03-01T00:00:00Z
full work available at URL: https://eudml.org/doc/135191
review text: If \(A\) is a subset of a semigroup \(S\), we denote by \(\langle A\rangle\) the subsemigroup generated by \(A\). The rank of a finite semigroup \(S\) is defined by \(\text{rank }S=\min\{| A|: A \subseteq S, \langle A\rangle = S\}\) and if it is generated by its set \(E\) of idempotents, its idempotent rank is defined by \(\text{idrank }S = \min\{| A|: A \subseteq E, \langle A\rangle = S\}\). Now let \(X_ n = \{1,2,\dots,n\}\), let \({\mathcal P\mathcal O}_ n\) denote the semigroup of all partial order preserving transformations on \(X_ n\), denote by \({\mathcal O}_ n\) the subsemigroup of \({\mathcal P\mathcal O}_ n\) consisting of all those transformations whose domains are \(X\) and let \({\mathcal S\mathcal P\mathcal O}_ n = {\mathcal P\mathcal O}_ n\setminus{\mathcal O}_ n\). The authors obtain a number of results about the ranks and idempotent ranks of these semigroups. For example, they show that if \(n \geq 2\), then \(\text{rank }{\mathcal O}_ n = n\) and \(\text{idrank }{\mathcal O}_ n = 2n-2\). They show further that \(\text{rank }{\mathcal P\mathcal O}_ n=2n-1\), \(\text{idrank }{\mathcal P\mathcal O_ n} = 3n-2\) and \(|{\mathcal P\mathcal O}_ n| + 1\) is the coefficient of \(x^ n\) in the Maclaurin expansion of the function \((1+x)^ n(1-x)^{- n}\). Finally, they show that \(\text{rank }{\mathcal S\mathcal P\mathcal O}_ n = 2n - 2\). Since \({\mathcal S\mathcal P\mathcal O}_ n\) is not generated by idempotents, the question of its idempotent rank does not arise. It should be mentioned that the manuscript somehow became incorrectly paginated in the process of getting into print. Pages 275 and 276 were interchanged.
reviewed by: jun. Kenneth D. Magill
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 20M05
zbMATH DE Number: 126128
zbMATH Keywords: partial order preserving transformations
zbMATH Keywords: ranks
zbMATH Keywords: idempotent ranks
zbMATH Keywords: Maclaurin expansion
MaRDI profile type: Publication
cites work: Idempotents in partial transformation semigroups
cites work: On the ranks of certain finite semigroups of transformations
cites work: Products of idempotents in certain semigroups of transformations
cites work: Q4126563
cites work: Idempotent rank in finite full transformation semigroups
OpenAlex ID: W2052621380
description: scientific article; zbMATH DE number 126128