Lattices of theories in languages without equality
type: Item
instance of: scholarly article
title: Lattices of theories in languages without equality
zbMATH Open document ID: 1278.08003
DOI: 10.1215/00294527-1960470
author: J. B. Nation
published in: Notre Dame Journal of Formal Logic
publication date: 2013-04-25T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ndjfl/1361454973
review text: In [\textit{K. Adaricheva} and \textit{J. B. Nation}, Int. J. Algebra Comput. 22, No. 7, Paper No. 1250065, 27 p. (2012; Zbl 1284.08018)] it was shown that every lattice of quasi-equational theories is isomorphic to the congruence lattice of some semilattice with operators. In [\textit{K. Adaricheva} and \textit{J. B. Nation}, ibid. 22, No. 7, Paper No. 1250066, 19 p. (2012; Zbl 1284.08019)] a partial converse was proved: under some additional restrictions on the semilattice and its operators, its congruence lattice can be represented as a lattice of quasi-equational theories. Here, it is shown that without these restrictions, the congruence lattice of a semilattice with operators is isomorphic to a lattice of implicational theories in a language which does not contain equality. The paper is concluded with an overview of other results and further questions. There is also a longer version of the paper available on the author's website. There are additional sections on models and satisfaction in both atomic and implicational theories.
reviewed by: Anna Zamojska-Dzienio
Mathematics Subject Classification ID: 08C15
Mathematics Subject Classification ID: 03G10
Mathematics Subject Classification ID: 06A12
Mathematics Subject Classification ID: 06B10
Mathematics Subject Classification ID: 06B15
Mathematics Subject Classification ID: 08B05
Mathematics Subject Classification ID: 08B15
zbMATH DE Number: 6157720
zbMATH Keywords: quasivariety
zbMATH Keywords: quasi-equational theory
zbMATH Keywords: congruence lattice
zbMATH Keywords: semilattice with operators
zbMATH Keywords: representation
zbMATH Keywords: lattice of implicational theories
MaRDI profile type: Publication
OpenAlex ID: W1988458761
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description: scientific article; zbMATH DE number 6157720