Characterization of subdirectly irreducible heyting algebras with negative tense operators

Ma and Li (2023, Stud. Log., 111, 21–56) established Intuitionistaic Propositional Logic with Galois negations (IGN). These logics may be regarded as the ordered duals of Ewald’s intuitionistic tense logic IKt, with Galois negations acting as the ordered duals of residuated pairs of tense operators. In Almiñana et al. (2023, Stud. Log., 111, 1015–1036), we introduced the notion of negative tense operators on Heyting algebras and defined the variety of tense H-algebras. This new variety provides the algebraic semantics for IGN. In this paper, we extend our investigation of tense H-algebras, with a particular focus on their topological properties. In particular, we provide a topological characterization of subdirectly irreducible tense H-algebras.

T-rough symmetric Heyting algebras with tense operators

Ewald (J Symbolic Logic 51(1):166---179, 1986) considered tense operators $$G$$ G , $$H$$ H , $$F$$ F and $$P$$ P on intuitionistic propositional calculus and constructed an intuitionistic tense logic system called IKt. The aim of this paper is to give an algebraic axiomatization of the IKt system. We will also show that the algebraic axiomatization given by Chajda (Cent Eur J Math 9(5):1185---1191, 2011) of the tense operators $$P$$ P and $$F$$ F in intuitionistic logic is not in accordance with the Halmos definition of existential quantifiers. In this paper, we will study the IKt variety of IKt-algebras. First, we will introduce some examples and we will prove some properties. Next, we will prove that the IKt system has IKt-algebras as algebraic counterpart. We will also describe a discrete duality for IKt-algebras bearing in mind the results indicated by Orlowska and Rewitzky (Fundam Inform 81(1---3):275---295, 2007) for Heyting algebras. We will also get a general construction of tense operators on a complete Heyting algebra, which is a power lattice via the so-called Heyting frame. Finally, we will introduce the notion of tense deductive system which allowed us both to determine the congruence lattice in an IKt-algebra and to characterize simple and subdirectly irreducible algebras of the IKt variety.

On Heyting Algebras with Negative Tense Operators