Communications of the
Korean Mathematical Society CKMS

In this article, we continue the study of tense symmetric Heyting algebras (or $TSH$-algebras). These algebras constitute a generalization of tense algebras. In particular, we describe a discrete duality for $TSH$-algebras bearing in mind the results indicated by Or\l owska and Rewitzky in [E. Or\l owska and I. Rewitzky, $Discrete$ $dualities$ $for$ $Heyting$ $algebras$ $with$ $operators$, Fund. Inform. $\bf 81$ (2007), no. 1-3, 275--295] for Heyting algebras. In addition, we introduce a propositional calculus and prove this calculus has $TSH$-algebras as algebraic counterpart. Finally, the duality mentioned above allowed us to show the completeness theorem for this calculus.

Keywords: symmetric Heyting algebras, tense operators, frames, discrete duality

MSC numbers: Primary 03G25, 06D50, 03B44