On the implicative‐infimum subreducts of weak Heyting algebras

• Published in: Mathematical logic quarterly Vol. 70; no. 2; pp. 178 - 196
• Main Authors: Celani, Sergio, San Martín, Hernán J.
• Format: Journal Article
• Language: English
• Published: Berlin Wiley Subscription Services, Inc 01.05.2024
• Subjects: Algebra; Infimum; Lattices (mathematics)
• ISSN: 0942-5616; 1521-3870
• DOI: 10.1002/malq.202300021

The variety of weak Heyting algebras was introduced in 2005 by Celani and Jansana. This corresponds to the strict implication fragment of the normal modal logic K$K$ which is also known as the subintuitionistic local consequence of the class of all Kripke models. Subresiduated lattices are a generalization of Heyting algebras and particular cases of weak Heyting algebras. They were introduced during the 1970s by Epstein and Horn as an algebraic counterpart of some logics with strong implication previously studied by Lewy and Hacking. In this paper we study the class of implicative‐infimum subreducts of weak Heyting algebras. In particular, we prove that this class is a variety by giving an equational base for it. We also present a topological duality for the algebraic category whose objects are the implicative‐infimum subreducts of subresiduated lattices.