Algebraically closed algebras in certain small congruence distributive varieties
type: Item
instance of: scholarly article
title: Algebraically closed algebras in certain small congruence distributive varieties
zbMATH Open document ID: 1198.08001
published in: Algebra Universalis
publication date: 2009-12-14T00:00:00Z
review text: According to \textit{J. Schmid} [Z. Math. Logik Grundlagen Math. 25, 525--530 (1979; Zbl 0425.06007)], a distributive lattice is algebraically closed iff it is relatively complemented. This characterization is an easy consequence of the much more general main theorem of the present paper: If \(\mathcal V\) is a finitely generated congruence-distributive variety such that any direct product of algebraically closed algebras from \(\mathcal V\) is also algebraically closed, then the algebraically closed algebras in \(\mathcal V\) are precisely the updirected unions of finite direct products of maximal subdirectly irreducibles from \(\mathcal V\); moreover, the class of all algebraically closed algebras in \(\mathcal V\) is elementary, to wit, definable by universal-existential Horn sentences (which is obvious in the example cited above).
reviewed by: Manfred Armbrust
Mathematics Subject Classification ID: 08B10
Mathematics Subject Classification ID: 08B25
Mathematics Subject Classification ID: 08C10
zbMATH DE Number: 5646232
zbMATH Keywords: algebraically closed algebra
zbMATH Keywords: absolute retract
zbMATH Keywords: amalgamation
zbMATH Keywords: congruence distributivity
author: Peter Ouwehand
MaRDI profile type: Publication
full work available at URL: s00012-009-0015-1
OpenAlex ID: W2032686865
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DOI: 10.1007/S00012-009-0015-1
description: scientific article; zbMATH DE number 5646232