Model completions and r-Heyting categories
type: Item
instance of: scholarly article
title: Model completions and r-Heyting categories
zbMATH Open document ID: 0888.03024
DOI: 10.1016/S0168-0072(97)00012-2
author: Silvio Ghilardi
author: Marek Zawadowski
published in: Annals of Pure and Applied Logic
publication date: 1998-05-11T00:00:00Z
review text: Using a syntactic version of Pitts' Theorem, the authors show in a first section that the first-order theory of Heyting algebras admits a model completion and that the opposite of the category of finitely represented Heyting algebras is a Heyting category. Following a brief recall of some notions concerning finitely represented algebras and the consideration of \(r\)-Heyting categories and their basic properties, a generalisation of the observation about Heyting algebras is proved in the next sections. Let \(T\) be an equational theory. Then the main result says that, under the assumption that the posets of regular subobjects of the opposite \({\mathbf T}\) of the category of locally presented \(T\)-algebras carry a Heyting algebra structure which is pullback-stable, \(T\) admits a model completion iff \({\mathbf T}\) is an \(r\)-Heyting category. The assumption above is equivalent to a conjunction of well-known notions from universal algebra. In an appendix it is shown that there are exactly 8 varieties of Heyting algebras whose theory admits a model completion.
Mathematics Subject Classification ID: 03C65
Mathematics Subject Classification ID: 03G30
Mathematics Subject Classification ID: 03C10
Mathematics Subject Classification ID: 08C05
Mathematics Subject Classification ID: 06D20
Mathematics Subject Classification ID: 06E25
Mathematics Subject Classification ID: 08C10
zbMATH DE Number: 1084238
zbMATH Keywords: intuitionistic propositional calculus
zbMATH Keywords: categorical structure
zbMATH Keywords: \(r\)-regular category
zbMATH Keywords: quantifier-elemination
zbMATH Keywords: category of locally presented \(T\)-algebras
zbMATH Keywords: first-order theory of Heyting algebras
zbMATH Keywords: model completion
zbMATH Keywords: category of finitely represented Heyting algebras
zbMATH Keywords: Heyting category
zbMATH Keywords: equational theory
zbMATH Keywords: varieties
reviewed by: Hans-Jürgen Vogel
MaRDI profile type: Publication
cites work: Representation of categories
cites work: On the structure of varieties with equationally definable principal congruences. I
cites work: A finite basis theorem for quasivarieties
cites work: Model theory.
cites work: Semantical investigations in Heyting's intuitionistic logic
cites work: Lokal präsentierbare Kategorien. (Locally presentable categories)
cites work: A sheaf representation and duality for finitely presented Heyting algebras
cites work: Varieties with equationally definable principal congruences
cites work: Completeness results for intuitionistic and modal logic in a categorical setting
cites work: Q4187817
cites work: Q3363966
cites work: Conceptual completeness for first-order intuitionistic logic: An application of categorical logic
cites work: On an interpretation of second order quantification in first order intuitionistic propositional logic
cites work: Model-companions and definability in existentially complete structures
description: scientific article; zbMATH DE number 1084238