| dc.description.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] | it_IT |
