On propositional logic semirings

Abstract: Propositional logic serves as a fundamental cornerstone in mathematical logic. This paper delves into a semiring characterization of propositional logic, employing the Gr\"oebner-Shirshov basis theory to furnish an algebraic framework for deduction and proof grounded in atoms of propositional logic. The result is an algebraic approach to proving propositions in propositional logic. To illustrate the effectiveness and constraints of this method, we conclude with several specific examples.