Triangular irreducibility of congruences in quasivarieties
type: Item
instance of: scholarly article
title: Triangular irreducibility of congruences in quasivarieties
zbMATH Open document ID: 1294.08003
author: Janusz Czelakowski
published in: Algebra Universalis
publication date: 2014-05-23T00:00:00Z
review text: The paper under review deals with forms of irreducibility as well as equational definability of relative congruences in quasivarieties. For a natural number \(m\geq 3\), a quasivariety \(\mathbf{Q}\) and \(\mathbf{A}\in \mathbf{Q}\), a \(\mathbf{Q}\)-congruence \(\Phi\) (i.e. \(\mathbf{A}/\Phi\in \mathbf{Q}\)) is said to be \textit{m-triangularily irreducible} in the lattice \(\text{Con}_\mathbf{Q}(\mathbf A)\) if for every sequence \(a_1, \dots , a_m\) of elements of \(\mathbf{A}\) (repetitions allowed), if \(\bigcap_{1\leq i<j\leq m}(\Theta_{\mathbf{Q}}^{\mathbf{A}}(a_i, a_j)+_{\mathbf{Q}}\Phi)=\Phi\) then \( (a_i,a_j)\in \Phi\) for some \(i<j\). Here, \(\Theta^{\mathbf A}_{\mathbf Q}(a_i,a_j)\) are certain \(\mathbf Q\)-congruences. \(\mathbf{A}\in \mathbf{Q}\) is said to be \(m\)-triangularily irreducible in \(\mathbf{Q}\) if \(\mathbf{0}_\mathbf{A}\) is \(m\)-triangularily meet irreducible in the lattice \(\text{Con}_{\mathbf{Q}}(\mathbf{A})\). It is easy to see that \(\mathbf{Q}_{\mathrm{RFSI}}\subseteq \mathbf{Q}_{m\mathrm{-TRI}}\). For \(m>3\), \(\mathbf{A}\in \mathbf{Q}_{m\mathrm{-TRI}}\) iff \(|A|<m\) or \(\mathbf{A}\in \mathbf{Q}_{\mathrm{RFSI}}\). It follows that if \(\mathbf{A}\in \mathbf{Q}\) then \(\mathbf{A}\) is a subdirect product of members of \(\mathbf{Q}_{m\mathrm{-TRI}}\). The author gives various characterization of finitely generated quasi-varieties via \(\mathbf{Q}\)-congruences of the form \(\bigcap_{1\leq i<j\leq m}\Theta_{\mathbf{Q}}^{\mathbf{A}}(a_i, a_j)\). Here is one of them: \(\mathbf{Q}\) is generated by a finite number of algebras of cardinality less than \(m\) iff for every \(\mathbf{A}\in \mathbf{Q}\) and any sequence \(a_1, \dots , a_m\) of elements of \(\mathbf{A}\) (possibly with repetitions), it is the case that \(\bigcap_{1\leq i<j\leq m}\Theta_{\mathbf{Q}}^{\mathbf{A}}(a_i, a_j)=\mathbf{0}_\mathbf{A}\). Let \(m\geq 3\) and \(\Lambda(x_1, \dots x_m, \underline{u})\) a set of equations in variables \(x_1, \dots , x_m\) and po…
Mathematics Subject Classification ID: 08A30
Mathematics Subject Classification ID: 08B05
Mathematics Subject Classification ID: 08B26
Mathematics Subject Classification ID: 08C15
zbMATH DE Number: 6297987
zbMATH Keywords: quasivariety
zbMATH Keywords: relative congruence
zbMATH Keywords: \(m\)-triangularily meet-irreducible relative congruence
zbMATH Keywords: equationally definable \(m\)-triangular meets of (relatively) principal congruences
zbMATH Keywords: finitely generated quasivariety
reviewed by: Hernando Gaitán
MaRDI profile type: Publication
full work available at URL: s00012-014-0274-3
OpenAlex ID: W2057711431
Wikidata QID: Q59397631
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DOI: 10.1007/S00012-014-0274-3
description: scientific article