Splitting p-algebras
type: Item
instance of: scholarly article
title: Splitting p-algebras
zbMATH Open document ID: 0547.06007
DOI: 10.1007/BF01198528
published in: Algebra Universalis
publication date: 1984-01-01T00:00:00Z
review text: A pair (\({\mathcal K}_ 0,{\mathcal K}_ 1)\) of equational subclasses of an equational class \({\mathcal K}\) is splitting in \({\mathcal K}\), if, for every equational subclass \({\mathcal K}_ 2\) of \({\mathcal K}\), either \({\mathcal K}_ 2\subseteq {\mathcal K}_ 0\) or \({\mathcal K}_ 1\subseteq {\mathcal K}_ 2\) in \({\mathcal K}\). If (\({\mathcal K}_ 0,{\mathcal K}_ 1)\) is a splitting pair in \({\mathcal K}\), \({\mathcal K}_ 1\) is generated by a subdirectly irreducible algebra A, called a splitting algebra for \({\mathcal K}\), and \({\mathcal K}_ 0\) is the largest subvariety of \({\mathcal K}\) not containing A, called the conjugate variety. It is generated by a single identity, called the conjugate identity. (This notion is due to \textit{R. McKenzie} [Trans. Am. Math. Soc. 174, 1-43 (1973; Zbl 0265.08006)]; for other structures, see e.g. \textit{A. Day} [Algebra Univers. 5, 153-162 (1975; Zbl 0324.08004)], and \textit{B. Jónsson} [ibid. 15, 273-298 (1982; Zbl 0545.08009)].) In the article under review, splitting algebras in various classes of p- algebras are studied. For \(\omega \geq n\geq 0\), \({\mathcal P}_ n\) is the equational class of p-algebras satisfying Lee's identity \(L_ n\), cf. \textit{K. Lee} [Can. J. Math. 22, 881-891 (1970; Zbl 0244.06009)]. From the numerous interesting results of this article, we state just a few, the choice being due entirely to the reviewer's taste. 1) Every splitting algebra in \({\mathcal P}_ n\) is finite. 2) Every equational subclass of \({\mathcal P}_ n\) generated by a (finite) subdirectly irreducible algebra is (completely) join irreducible in the lattice of subvarieties of \({\mathcal P}_ n\). 3) Let \(n\in\{0,\omega\}\), and L be a finite subdirectly irreducible algebra in \({\mathcal P}_ n\). Then the following are equivalent: i) L is splitting in \({\mathcal P}_ n\), ii) L is a bounded homomorphic image of a free algebra in \({\mathcal P}_ n\) with finitely many generators. iii) Every homomorphism from a…
Mathematics Subject Classification ID: 06D15
Mathematics Subject Classification ID: 08B15
Mathematics Subject Classification ID: 06B20
Mathematics Subject Classification ID: 06B25
zbMATH DE Number: 3871429
zbMATH Keywords: equational class
zbMATH Keywords: subdirectly irreducible algebra
zbMATH Keywords: conjugate variety
zbMATH Keywords: splitting algebras
zbMATH Keywords: classes of p-algebras
zbMATH Keywords: Lee's identity
zbMATH Keywords: lattice of subvarieties
zbMATH Keywords: free algebra
zbMATH Keywords: finitely generated algebra
zbMATH Keywords: conjugate identities
author: Tibor Katriňák
MaRDI profile type: Publication
cites work: A Simple Solution to the Word Problem for Lattices
cites work: Splitting algebras and a weak notion of projectivity
cites work: Free Lattices Generated by Partially Ordered Sets and Preserving Bounds
cites work: Q3875978
cites work: Algebras Whose Congruence Lattices are Distributive.
cites work: Primitive Klassen von modularen S-Algebren.
cites work: Subdirectly irreducible modular p-algebras
cites work: The cardinality of the lattice of all equational classes of p-algebras
cites work: Subdirectly irreducible p-algebras
cites work: Projective lattices and bounded homomorphisms
cites work: The Structure of Pseudocomplemented Distributive Lattices. I: Subdirect Decomposition
cites work: Equational classes of lattices with pseudocomplementation
cites work: Equational Classes of Distributive Pseudo-Complemented Lattices
cites work: Equational Bases and Nonmodular Lattice Varieties
cites work: Q5799454
full work available at URL: https://doi.org/10.1007/bf01198528
OpenAlex ID: W4253762535
description: scientific article; zbMATH DE number 3871429