http://hdl.handle.net/2099/2063 "MV-algebras (part II)" 2013-05-25T11:22:02Z http://hdl.handle.net/2099/3675 Title: Editorial [Special issue: Second issue devoted to MV-algebras] Authors: Sessa, Salvatore 1997-01-01T00:00:00Z http://hdl.handle.net/2099/3486 Title: Orthogonal decompositions of MV-spaces Authors: Belluce, L.P.; Sessa, Salvatore Abstract: A maximal disjoint subset $S$ of an $MV$-algebra $A$ is a basis iff $\{x \in A : x \leq a \}$ is a linearly ordered subset of $A$ for all $a \in S$. Let $\Spec A$ be the set of the prime ideals of $A$ with the usual spectral topology. A decomposition $\Spec A = \cup_{i \in I} T_{i} \cup X$ is said to be orthogonal iff each $T_{i}$ is compact open and $S = \{a_{i}\}_{i\in I}$ is a maximal disjoint subset. We prove that this decomposition is unrefinable (i.e. no $T_{i} = \Theta \cap Y$ with $\Theta$ open, $\Theta \cap Y = \emptyset$, int $Y = \emptyset$) iff $S$ is a basis. Many results are established for semisimple $MV$-algebras, which are the algebraic counterpart of Bold fuzzy set theory. 1997-01-01T00:00:00Z http://hdl.handle.net/2099/3485 Title: Axiomatizing quantum MV-algebras Authors: Giuntini, Roberto Abstract: We introduce the notion of {\it p-ideal\/} of a QMV-algebra and we prove that the class of all $p$-ideals of a QMV-algebra $\C M$ is in one-to-one correspondence with the class of all congruence relations of $\C M$. 1997-01-01T00:00:00Z http://hdl.handle.net/2099/3484 Title: Convergence in MV-algebras Authors: Georgescu, George; Liguori, Fortuna; Martini, Giulia Abstract: $MV$-algebras were introduced in 1958 by Chang and they are models of Lukasiewicz infinite-valued logic. Chang gives a correspondence between the category of linearly ordered $MV$-algebras and the category of linearly ordered abelian $\ell$-groups. Mundici extended this result showing a categorical equivalence between the category of the $MV$-algebras and the category of the abelian $\ell$-groups with strong unit. In this paper, starting from some definitions and results in abelian $\ell$-groups, we shall study the convergent sequences and the Cauchy sequences in an $MV$-algebra. The main result is the construction of the Cauchy completion $A^{*}$ of an $MV$-algebra $A$. It is proved that a complete $MV$-algebra is also Cauchy complete. Additional results on atomic and complete $MV$-algebras are also given. 1997-01-01T00:00:00Z http://hdl.handle.net/2099/3483 Title: Maximal MV-algebras Authors: Filipoiu, Alexandru; Georgescu, George; Lettieri, Ada Abstract: In this paper we define maximal $MV$-algebras, a concept similar to the maximal rings and maximal distributive lattices. We prove that any maximal $MV$-algebra is semilocal, then we characterize a maximal $MV$-algebras as finite direct product of local maximal $MV$-algebras. 1997-01-01T00:00:00Z http://hdl.handle.net/2099/3482 Title: Representation of a Boolean algebra by its triangular norms Authors: Ray, Suryansu Abstract: Given a complete and atomic Boolean algebra $B$, there exists a family $\tau_{\gamma}$ of triangular norms on $B$ such that, under the partial ordering of triangular norms, $\tau_{\gamma}$ is a Boolean algebra isomorphic to $B$, where $\gamma$ is the set of all atoms in $B$. In other words, as we have shown in this note, every complete and atomic Boolean algebra can be represented by its own triangular norms. What we have not shown in this paper is our belief that $\tau_{\gamma}$ is not unique for $B$ and that, for such a representation, $B$ needs neither to be complete, nor to be atomic. 1997-01-01T00:00:00Z