On \(n \times m\)-valued Łukasiewicz-Moisil algebras
type: Item
instance of: scholarly article
title: On \(n \times m\)-valued Łukasiewicz-Moisil algebras
zbMATH Open document ID: 1155.06009
author: Claudia A. Sanza
published in: Central European Journal of Mathematics
publication date: 2008-09-12T00:00:00Z
review text: \(n\times m\)-valued \(\L\)ukasiewicz algebras with negation (called \(\text{LM}_{n\times m}\)-algebras) were introduced and investigated by the author of this article and constitute a non-trivial generalization of \(n\)-valued \(\L\)ukasiewicz-Moisil algebras. This paper continues the study of these algebras, more precisely, the author describes a topological duality for these algebras (the dual of the category of \(\text{LM}_{n\times m}\)-algebras is the category \(\mathbf{lm}_{n\times m}\) of \(\text{lm}_{n\times m}\)-spaces and \(\text{lm}_{n\times m}\)-functions) and characterizes the congruences on these algebras by means of special subsets of the associated spaces. In particular, the author determines which of these subsets correspond to principal congruences. Also, the author proves that the variety of \(n\times m\)-valued \(\L\)ukasiewicz-Moisil algebras is a discriminator variety (if it has a polynomial \(p\) that coincides with the ternary discriminator function on each subdirectly irreducible member of the variety) and, as a consequence, she obtains certain properties of the congruences, such as: every principal congruence is a factor congruence, principal congruences form a sublattice in the congruence lattice, each compact congruence is a principal congruence, the congruences are regular, normal, filtral, and the category \(\mathbf{LM}_{n\times m}\) has the congruence extension property. Finally, she computes the number of congruences of such finite algebras: if \(L\) is a finite \(\text{LM}_{n\times m}\)-algebra and \(A\) is the set of all atoms of its center, then \(|\text{Con}_{\text{LM}_{n\times m}}(L)| =2^{A}\).
reviewed by: Florentina Chirteş
Mathematics Subject Classification ID: 06D30
Mathematics Subject Classification ID: 03G20
Mathematics Subject Classification ID: 06D50
zbMATH DE Number: 5343479
zbMATH Keywords: Priestley spaces
zbMATH Keywords: discriminator varieties
zbMATH Keywords: congruences
MaRDI profile type: Publication
full work available at URL: s11533-008-0035-7
OpenAlex ID: W1978203612
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DOI: 10.2478/S11533-008-0035-7
description: scientific article; zbMATH DE number 5343479