DSpace Collection: "MV-algebras (part II)"
http://hdl.handle.net/2099/2063
"MV-algebras (part II)"Thu, 18 Sep 2014 22:01:51 GMT2014-09-18T22:01:51Zwebmaster.bupc@upc.eduUniversitat Politècnica de Catalunya. Servei de Biblioteques i DocumentaciónoEditorial [Special issue: Second issue devoted to MV-algebras]
http://hdl.handle.net/2099/3675
Title: Editorial [Special issue: Second issue devoted to MV-algebras]
Authors: Sessa, SalvatoreWed, 01 Jan 1997 00:00:00 GMThttp://hdl.handle.net/2099/36751997-01-01T00:00:00ZSessa, SalvatorenoOrthogonal decompositions of MV-spaces
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.Wed, 01 Jan 1997 00:00:00 GMThttp://hdl.handle.net/2099/34861997-01-01T00:00:00ZBelluce, L.P.; Sessa, SalvatorenoMV-algebra, Orthogonal decomposition, Basis, Closed ideal, Semisimple MV-algebra, Complete MV-algebraA 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.Axiomatizing quantum MV-algebras
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 $\C M$.Wed, 01 Jan 1997 00:00:00 GMThttp://hdl.handle.net/2099/34851997-01-01T00:00:00ZGiuntini, RobertonoQMV-algebra, MV-algebra, Quantun logicWe 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 $\C M$.Convergence in MV-algebras
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.Wed, 01 Jan 1997 00:00:00 GMThttp://hdl.handle.net/2099/34841997-01-01T00:00:00ZGeorgescu, George; Liguori, Fortuna; Martini, GiulianoComplete MV-algebras$MV$-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.Maximal MV-algebras
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.Wed, 01 Jan 1997 00:00:00 GMThttp://hdl.handle.net/2099/34831997-01-01T00:00:00ZFilipoiu, Alexandru; Georgescu, George; Lettieri, AdanoMV-algebrasIn 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.Representation of a Boolean algebra by its triangular norms
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.Wed, 01 Jan 1997 00:00:00 GMThttp://hdl.handle.net/2099/34821997-01-01T00:00:00ZRay, SuryansunoTriangular norms, Complete and atomic Boolean algebra, Fuzzy groupsGiven 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.