Semiring and semimodule issues in MV-algebras

In this paper we propose a semiring-theoretic approach to MV-algebras based on the connection between such algebras and idempotent semirings - such an approach naturally imposing the introduction and study of a suitable corresponding class of semimodules, called MV-semimodules. We present several results addressed toward a semiring theory for MV-algebras. In particular we show a representation of MV-algebras as a subsemiring of the endomorphism semiring of a semilattice, the construction of the Grothendieck group of a semiring and its functorial nature, and the effect of Mundici categorical equivalence between MV-algebras and lattice-ordered Abelian groups with a distinguished strong order unit upon the relationship between MV-semimodules and semimodules over idempotent semifields.Comment: This version contains some corrections to some results at the end of Section

Modal operators on bounded residuated l\rm l-monoids

summary:Bounded residuated lattice ordered monoids (${\rm R\ell}$-monoids) form a class of algebras which contains the class of Heyting algebras, i.e.\ algebras of the propositional intuitionistic logic, as well as the classes of algebras of important propositional fuzzy logics such as pseudo $\mathop{\rm MV}$-algebras (or, equivalently, $\mathop{\rm GMV}$-algebras) and pseudo $\mathop{\rm BL}$-algebras (and so, particularly, $\mathop{\rm MV}$-algebras and $\mathop{\rm BL}$-algebras). Modal operators on Heyting algebras were studied by Macnab (1981), on $\mathop{\rm MV}$-algebras were studied by Harlenderová and Rachůnek (2006) and on bounded commutative ${\rm R\ell}$-monoids in our paper which will apear in Math. Slovaca. Now we generalize modal operators to bounded ${\rm R\ell}$-monoids which need not be commutative and investigate their properties also for further derived algebras