Functional monadic Heyting algebras.
type: Item
instance of: scholarly article
title: Functional monadic Heyting algebras.
zbMATH Open document ID: 1062.06017
author: Guram Bezhanishvili
author: John Harding
published in: Algebra Universalis
publication date: 2005-04-19T00:00:00Z
review text: A Heyting algebra is a bounded distributive lattice with the operation of relative pseudocomplementation. A monadic Heyting algebra is a Heyting algebra equipped with two unary operations \(\Delta \) and \(\nabla \) satisfying the conditions (i) \(\Delta \nabla a = \nabla a\) (ii) \(\nabla \Delta a = \Delta a\) (iii) \(\nabla (\nabla a\wedge b) = \nabla a \wedge \nabla b\). It forms an algebraic counterpart of intuitionistic predicate calculus where \(\Delta \) or \(\nabla \) are required as the intuitionistic quantifiers \(\forall \) or \(\exists \), respectively. For a complete Heyting algebra \(H\) and a set \(X\), define operations \(\Delta , \nabla \) on the Heyting algebra \(H^X\) of all functions \(f: X\to H\) by setting \[ (\Delta f)(x) = \bigwedge \{ f(y);\;y\in X \} \quad \text{and} \quad (\nabla f)(x) = \bigvee \{ f(y);\;y\in X \}. \] The resulting algebra is called a functional Heyting algebra. It was a problem of A. Monteiro and O. Varsavsky whether every monadic Heyting algebra is isomorphic to a functional monadic Heyting algebra. The authors present a new method for solving this problem in the positive.
reviewed by: Ivan Chajda
Mathematics Subject Classification ID: 06D20
Mathematics Subject Classification ID: 03G25
zbMATH DE Number: 2158793
zbMATH Keywords: monadic Heyting algebra
zbMATH Keywords: amalgamation
zbMATH Keywords: algebras of intuitionistic logic
MaRDI profile type: Publication
full work available at URL: s00012-002-8202-3
OpenAlex ID: W2029285695
DOI: 10.1007/S00012-002-8202-3
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description: scientific article; zbMATH DE number 2158793