Hilbert Algebras with a Modal Operator $${\Diamond}$$ ◊

Abstract

A Hilbert algebra with supremum is a Hilbert algebra where the associated order is a join-semilattice. This class of algebras is a variety and was studied in Celani and Montangie (2012). In this paper we shall introduce and study the variety of $${H_{\Diamond}^{\vee}}$$H??-algebras, which are Hilbert algebras with supremum endowed with a modal operator $${\Diamond}$$?. We give a topological representation for these algebras using the topological spectral-like representation for Hilbert algebras with supremum given in Celani and Montangie (2012). We will consider some particular varieties of $${H_{\Diamond}^{\vee}}$$H??-algebras. These varieties are the algebraic counterpart of extensions of the implicative fragment of the intuitionistic modal logic $${\mathbf{IntK}_{\Diamond}}$$IntK?. We also determine the congruences of $${H_{\Diamond}^{\vee}}$$H??-algebras in terms of certain closed subsets of the associated space, and in terms of a particular class of deductive systems. These results enable us to characterize the simple and subdirectly irreducible $${H_{\Diamond}^{\vee }}$$H??-algebras.