Abstract 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.

Abstract In this paper, we introduce and study an alternative definition of tense operators on residuated lattices. We give a categorical equivalence for the class of tense ICRDL-algebras, which is motivated by an old construction by J. Kalman. The paper concludes with some applications regarding descriptions of congruences and a 2-contextual translation.

Abstract This paper focuses on semilattices with adjunctions (SLatas), which are semilattices with a greatest element enriched with a pair of adjoint maps. We develop a spectral-style duality for SLatas, building on prior topological dualities for monotone semilattices. As an application of this duality, we characterize SLata congruences through an adaptation of lower-Vietoris-type families. Furthermore, we investigate tense operators on semilattices with a greatest element, deriving a duality for this case by extending the results obtained for SLatas.

In 2007, tense Łukasiewicz–Moisil algebras were introduced by Diaconescu and Georgescu as an algebraic counterpart of tense n–valued Moisil logic. These algebras constitute a generalization of tense algebras. In this paper we describe a discrete duality for tense Łukasiewicz–Moisil algebras bearing in mind the results indicated by Dzik, Orłowska and van Alten in 2006, for De Morgan algebras.