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.

We obtain a duality between the category of locally unital finite MTL-algebras and the category of finite labeled trees. In addition we prove that certain poset products of MTL-algebras are in fact, sheaves of MTL-chains over Alexandrov spaces. Finally we give a concrete description for the studied poset products in terms of direct products and ordinal sums of finite MTL-algebras.