On tense bounded Hilbert algebras with supremum

Abstract Hilbert algebras with supreme were initially considered by A. V. Figallo, G. Ramón and S. Saad in 1998. In this article, we present and study the variety of tense $ H_{0}^ {\vee }$-algebras, which are bounded Hilbert algebras with supremum, endowed with the tense operators $ G $, $ H $, $ F $ and $ P $. We give the notion of a tense deductive system, and we prove that the lattice of the congruences of a tense $ H_{0}^ {\vee } $-algebra and the lattice of the tense deductive systems of it are isomorphic. We introduce a special type of topological spaces, called tense $ H_{0}^{\vee } $-spaces. We prove that the category of tense $H_{0}^ {\vee } $-algebras with semi-homomorphisms is naturally equivalent to the category of tense $ H_{0}^ {\vee } $-spaces with certain relations. We show that the lattice of the congruences of a tense $ H_{0}^ {\vee } $-algebra and the lattice of certain closed subsets of its associated tense $H_{0}^ {\vee }$-space are dually isomorphic. Moreover, we characterize by topological methods the subdirectly irreducible tense $H_{0}^ {\vee }$-algebras and particularly the simple tense $H_{0}^ {\vee }$-algebras.