MV-algebras, semirings and their applications

Abstract

The thesis is divided in two parts: the first part regards MV-semirings, involutive semirings and semimodules over them with particular attention to injective and projective semimodules; the second part of the thesis is focused on the tropical semiring and has the purpose to characterize the sets which arise as images of retractions that are nonexpansive with respect to a hemi-norm which plays a key role in tropical geometry. Semirings and semimodules, and their applications, arise in various branches of Mathematics, Computer Science, Physics, as well as in many other areas of modern science (see, for instance, [3]). MV-algebras arose in the literature as the algebraic semantics of ukasiewicz propositional logic, one of the longest-known many-valued logic. A connection between MV-algebras and a special category of additively idempotent semirings (called MV-semirings or ukasiewicz semirings) was rst observed in [1]. On the one hand, every MV-algebra has two semiring reducts isomorphic to each other by the involutive unary operation of MV-algebras (see, e.g., [2, Proposition 4.8]); on the other hand, the category of MV-semirings de ned in [2] is isomorphic to the one of MV-algebras. The term equivalence between MV-algebras and MV-semirings allows us to import results and techniques of semiring and semimodule theory in the study of MV-algebras as well to use properties and theorems regarding MV-algebras in the study of semimodules over MV-semirings. Indeed, as the theory of modules is an essential chapter of ring theory, so the theory of semimodules is a crucial aspect in semiring theory and two of the most important objects in semimodule theory are projective and injective semimodules. Although, in general, describing the structure of projective and injective semimodules seems to be a quite di cult task, we shall give a criterion for injectivity of semimodules over additively idempotent semirings which we shall use to describe the structure of injective semimodules over MVsemirings with an atomic Boolean center, i. e. the boolean elements of the MV-semiring form an atomic lattice... [edited by Author]