On tense bounded Hilbert algebras with supremum

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 H₀^-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 H₀^-algebra and the lattice of the tense deductive systems of it are isomorphic. We introduce a special type of topological spaces, called tense H₀-spaces. We prove that the category of tense H₀^-algebras with semi-homomorphisms is naturally equivalent to the category of tense H₀^-spaces with certain relations. We show that the lattice of the congruences of a tense H₀^-algebra and the lattice of certain closed subsets of its associated tense H₀^-space are dually isomorphic. Moreover, we characterize by topological methods the subdirectly irreducible tense H₀^-algebras and particularly the simple tense H₀^-algebras.