MVW-rigs

Abstract

In the context of the MV-algebras there is a class of them endowed with a product. A subclass of this class of MV-algebras has been studied by Dinola in [7]; the MV-algebras product or MVP. Thus for example the MV-algebra [0, 1] is closed for the usual product between real numbers. It is known that this product respects the usual order that in turn coincides with the natural order associated with this MV-algebra. Similarly, the algebra of continuous functions of [0, 1] n in [0, 1]. In the present work the class of MV-algebras with products is characterized in a wider context than that presented by Dinola. From the properties of the universal algebra found in the MV-algebras of closed continuous functions for products it will be shown that this general context is more convenient to work properties analogous to commutative algebra.As a result of this characterization a new algebraic structure is defined, which is an MV-algebra endowed with a product operation, which we will call MVW-rig (Weak-Rig Multivalued) because of its close relation with the rigs defined in [2]. This structure is defined with axioms of universal algebra, a good number of natural examples are presented in the MV-algebras environment and the first results concerning homomorphisms, ideals, quotients and subdirect products are established. In particular, his prime spectrum is studied which, with the co-zariski topology defined by Dubuc, Poveda in [8] is compact. Consequently, a good number of results analogous to the theory of commutative rings and rigs are presented, with which this theory maintains a close relation.