Monoids respecting \(n\)-chains of intervals
type: Item
instance of: scholarly article
title: Monoids respecting \(n\)-chains of intervals
zbMATH Open document ID: 0873.20040
author: Jorge Almeida
author: Peter M. Higgins
published in: Journal of Algebra
publication date: 1997-09-01T00:00:00Z
review text: Let \(\mathbb{O}\) (resp., \(\mathbb{P}\mathbb{O}\)) denote the monoid pseudovariety generated by all monoids of (partial) order-preserving mappings on a finite chain. Starting with these pseudovarieties, the authors construct two strict hierarchies \(\mathbb{O}\subset\mathbb{O}^2\subset\ldots\subset\mathbb{O}^n\subset\ldots\) and \(\mathbb{P}\mathbb{O}\subset\mathbb{P}\mathbb{O}^2\subset\ldots\subset\mathbb{P}\mathbb{O}^n\subset\ldots\) of aperiodic monoid pseudovarieties both giving as the union the pseudovariety of all finite aperiodic monoids. The members of the hierarchies are defined in terms of their generating transformation monoids that respect a certain chain of interval partitions of a finite chain. On the other hand, the \(n\)th member of each hierarchy can be obtained as the \(n\)th semidirect power of the first member (Theorem 4). The second hierarchy can be obtained from the first one by semidirectly multiplying through on the left by the pseudovariety of all finite semilattices (Theorem 8). This implies that the hierarchies intermingle in the following way: \(\mathbb{O}\subset\mathbb{P}\mathbb{O}\subset\mathbb{O}^2\subseteq\mathbb{P}\mathbb{O}^2\subseteq\ldots\subseteq\mathbb{O}^n\subseteq\mathbb{P}\mathbb{O}^n\subseteq\mathbb{O}^{n+1}\subseteq\ldots\) (Corollary 9). The question as to whether \(\mathbb{O}^n\neq\mathbb{P}\mathbb{O}^n\) for all \(n\) remains open. The authors also find, for each \(n\), a pseudoidentity valid in \(\mathbb{O}^n\) (Proposition 10) but not in the pseudovariety \(\mathbb{L}\) of all finite \(\mathcal L\)-trivial semigroups (Proposition 12) thus proving that \(\mathbb{L}\) belongs to no finite level of any of the two hierarchies.
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 08C15
Mathematics Subject Classification ID: 08B15
Mathematics Subject Classification ID: 08B25
Mathematics Subject Classification ID: 20M20
zbMATH DE Number: 1014037
zbMATH Keywords: partial order-preserving mappings of finite chains
zbMATH Keywords: aperiodic monoid pseudovarieties
zbMATH Keywords: semidirect products of pseudovarieties
zbMATH Keywords: pseudoidentities
zbMATH Keywords: generating transformation monoids
MaRDI profile type: Publication
full work available at URL: jabr.1997.6783
OpenAlex ID: W2031718907
cites work: The join of the pseudovarieties of \(\mathcal R\)-trivial and \(\mathcal L\)-trivial monoids
cites work: The Gap Between Partial and Full
cites work: Profinite categories and semidirect products
cites work: Defining Relations and the Algebraic Structure of the Group SL2 over Integral Hamilton Quaternions
cites work: Q4132170
cites work: Semigroups of order preserving mappings on a finite chain: A new class of divisors
cites work: DIVISORS OF SEMIGROUPS OF ORDER-PRESERVING MAPPINGS ON A FINITE CHAIN
cites work: Algebraic Theory of Machines. I. Prime Decomposition Theorem for Finite Semigroups and Machines
cites work: Q3769981
cites work: The Birkhoff theorem for finite algebras
cites work: Q4077455
cites work: On finite \(\mathcal J\)-trivial monoids
cites work: Partially ordered finite monoids and a theorem of I. Simon
cites work: The proof and the generalization of Higgins' theorem of divisors of semigroups of order-preserving mappings
DOI: 10.1006/JABR.1997.6783
description: scientific article; zbMATH DE number 1014037