Galois connections and tense operators on q-effect algebras

TL;DR

This paper explores the representation of tense operators on q-effect algebras, introducing q-effect algebras to better model temporal aspects in quantum and many-valued logics.

Contribution

It introduces q-effect algebras and solves the representation problem for q-tense operators on q-representable q-effect algebras.

Findings

Established a canonical construction of tense operators using time frames.

Proved the existence of time frames for q-tense operators on q-effect algebras.

Enhanced the modeling of temporal logic in quantum and many-valued systems.

Abstract

For effect algebras, the so-called tense operators were already introduced by Chajda and Paseka. They presented also a canonical construction of them using the notion of a time frame. Tense operators express the quantifiers "it is always going to be the case that" and "it has always been the case that" and hence enable us to express the dimension of time both in the logic of quantum mechanics and in the many-valued logic. A crucial problem concerning tense operators is their representation. Having an effect algebra with tense operators, we can ask if there exists a time frame such that each of these operators can be obtained by the canonical construction. To approximate physical real systems as best as possible, we introduce the notion of a q-effect algebra and we solve this problem for q-tense operators on q-representable q-Jauch-Piron q-effect algebras.