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An explicit categorical equivalence is defined between a proper subvariety of the class of $PMV$-algebras, as defined by Di Nola and Dvure$\check{c}$enskij, to be called $PMV_f$-algebras, and the category of semi-low $f_u$-rings. This categorical representation is done using the prime spectrum of the $MV$-algebras, through the equivalence between $MV$-algebras and $l_u$-groups established by Mundici, from the perspective of the Dubuc-Poveda approach, that extends the construction defined by Chang on chains. As a particular case, semi-low $f_u$-rings associated to Boolean algebras are characterized. Besides we show that class of $PMV_f$-algebras is coextensive.

In this article we develop a new version of the intuitionist existential graphs presented by Arnol Oostra [4]. The deductive rules presented in this article have the same meaning as those described in the work of Yuri Poveda [5], because the deductions according to the parity of the cuts are eliminated and are replaced by a finite set of recursive rules. This way, $ Alfa_I $ the existential graphs system for intuitional propositional logic follows the course of the deductive rules of the system $ Alfa_0 $ described by Poveda [5], and is equivalent to the intuitionistic propositional calculus. In this representation the $ Alfa_0 $ system is improved, there are a series of deductive rules of second degree incorporated that previously had not been considered and that allow a better management of deductions and finally from the ideas proposed by Van Dalen [6], a mixture is incorporated in the deduction techniques, the natural deductions of the Gentzen system are combined with new system rules $ Alfa_0 $ and $ Alfa_I $. The symbols proposed for the $Alfa_I$ representation relate open, closed and quasi-open sets of the usual topology of the plot with the intuitional propositional logic, usefull for approaching new problems in the representation of this logic from a more geometrical perspective.

In this paper, a new algebraic structure is defined, which is a new MV-algebra that has a product operation, we will call it MVW-rig (Multivalued-weak rig). This structure is defined with universal algebra axioms, it is presented with a good amount of natural examples in the MV-algebra environment and the first results having to do with ideal, quotients, homomorphisms and subdirect product are established. In particular, its prime spectrum is studied, that with the co-Zariski topology it is compact. Consequently, a good number of results that are analogous to the theory of commutative rings and rigs are presented with which this theory keeps a close relationship to., Comment: in English

In "A new proof of the completeness of the Lukasiewicz axioms"} (Transactions of the American Mathematical Society, 88) C.C. Chang proved that any totally ordered $MV$-algebra $A$ was isomorphic to the segment $A \cong \Gamma(A^*, u)$ of a totally ordered $l$-group with strong unit $A^*$. This was done by the simple intuitive idea of putting denumerable copies of $A$ on top of each other (indexed by the integers). Moreover, he also show that any such group $G$ can be recovered from its segment since $G \cong \Gamma(G, u)^*$, establishing an equivalence of categories. In "Interpretation of AF $C^*$-algebras in Lukasiewicz sentential calculus" (J. Funct. Anal. Vol. 65) D. Mundici extended this result to arbitrary $MV$-algebras and $l$-groups with strong unit. He takes the representation of $A$ as a sub-direct product of chains $A_i$, and observes that $A \overset {} {\hookrightarrow} \prod_i G_i$ where $G_i = A_i^*$. Then he let $A^*$ be the $l$-subgroup generated by $A$ inside $\prod_i G_i$. He proves that this idea works, and establish an equivalence of categories in a rather elaborate way by means of his concept of good sequences and its complicated arithmetics. In this note, essentially self-contained except for Chang's result, we give a simple proof of this equivalence taking advantage directly of the arithmetics of the the product $l$-group $\prod_i G_i$, avoiding entirely the notion of good sequence., Comment: 6 pages

In this note we exhibit a very simple proof of McNaughton Theorem, almost right out of the definitions, and at the same time we observe that this theorem does not depend of Chang's completeness theorem., Comment: 1 and 1/2 pages

In this paper we develop a general representation theory for mv-algebras. We furnish the appropriate categorical background to study this problem. Our guide line is the theory of classifying topoi of coherent extensions of universal algebra theories. Our main result corresponds, in the case of mv-algebras and mv-chains, to the representation of commutative rings with unit as rings of global sections of sheaves of local rings. \emph{We prove that any mv-algebra is isomorphic to the mv-algebra of all global sections of a sheaf of mv-chains on a compact topological space}. This result is intimately related to McNaughton's theorem, and we explain why our representation theorem can be viewed as a vast generalization of McNaughton's. On spite of the language utilized in this abstract, we wrote this paper in a way that, we hope, could be read without much acquaintance with either sheaf theory or mv-algebra theory., Comment: 35 pages