Algebraic approach to tense operators

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Propositional logics, both classical and non-classical, typically overlook the dimension of time, despite Aristotle's assertion that time significantly influences the truth values of propositions. He illustrated this with the statement "There will be a sea battle tomorrow," which cannot be assigned a truth value today. This limitation led him to conclude that two-valued logic fails to encompass the entirety of human reasoning. As logic evolved, particularly with the advent of computers in the 1940s and the rise of Artificial Intelligence, the ability to evaluate future truth values became crucial for controlling complex systems. This necessity spurred interest in temporal logic, which integrates time as a variable in propositional formulas. Arthur Prior introduced tense logic in the late 1950s, focusing on the interplay between tense and modality. This work includes modal operators alongside traditional truth-functional operators. The aim here is not to fully detail tense logic but to present an algebraic axiomatization of tense logic and its operators. While classical propositional logic was formalized by George Boole using Boolean algebras, subsequent developments have seen intuitionistic logic, many-valued logics, and fuzzy logic formalized through various algebraic structures. This monograph employs algebraic methods to axiomatize tense and modal operators, beginning with quantifiers as developed by P. Halmos and J. D. Rut