Another look on tense and related operators

TL;DR

This paper explores foundational algebraic structures related to tense and modal operators, introducing new semilattice constructions and functorial relationships with applications across mathematics.

Contribution

It presents three novel sup-semilattice constructions, including powerset operators, and demonstrates their role in forming adjoint functor pairs in algebraic and logical contexts.

Findings

Introduces three basic sup-semilattice constructions.

Establishes four covariant and two contravariant functors.

Provides simple examples of adjoint situations.

Abstract

Motivated by the classical work of Halmos on functional monadic Boolean algebras we derive three basic sup-semilattice constructions, among other things the so-called powersets and powerset operators. Such constructions are extremely useful and can be found in almost all branches of modern mathematics, including algebra, logic and topology. Our three constructions give rise to four covariant and two contravariant functors and constitute three adjoint situations we illustrate in simple examples.