Hilbert Algebras with a Modal Operator ◊

SERGIO ARTURO CELANI Y DANIELA MONTANGIE

STUDIA LOGICA

Springer

Año: 2015

0039-3215

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 cite{CelaniMontangie}. In this paper we shall introduce and study the variety of $H_{Diamond}^{ee}$-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 cite{CelaniMontangie}. We will consider some particular varieties of $H_{Diamond}^{ee}$-algebras. These varieties are the algebraic counterpart of extensions of the implicative fragment of the intuitionistic modal logic $mathbf{IntK}_{Diamond}$. We also determine the congruences of $H_{Diamond }^{ee}$-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 $H_{Diamond}^{ee }$-algebras.