Semisimple MV-algebras Research Articles

Two notions of MV-algebraic semisimplicity relative to fixed MV-chains

We initiate a study of two general concepts of semisimplicity for MV-algebras by replacing the standard MV-algebra with an arbitrary MV-chain . These generalised notions are called -semisimple MV-algebras and -semisimple MV-algebras. We obtain several of their characterisations and explore in more-depth the case of perfect MV-chains.

Ideals of semisimple MV-algebras and convergence along set-theoretic filters

In this paper, we investigate the connection between ideals of semisimple MV-algebras and set-theoretic filters. In particular, we obtain that there is a bijection between closed ideals (with respect to a specified closure operation) and all filters on some associated set X. Finally, we establish that the space of closed prime ideals is homeomorphic to a well-described space having the Stone-Čech compactification βX as a subspace.

Topological characterization of semisimple MV-algebras

In this paper, we will explain the reason that filter topological characterization of 2-divisible abelian lattice ordered groups cannot be easily translated into MV-algebras. Then we prove that a 2-divisible MV-algebra A is semisimple if and only if it is Hausdorff with respect to the filter topology induced by the lattice filter of the form F=⋃n∈NAa,n for any a∈A which exceeds zero. We also show that an MV-algebra is simple if and only if it is Hausdorff with respect to any filter topology. Furthermore, an MV-algebra A is semisimple if and only if for any lattice filter F, which satisfies F∩I=∅ for any proper MV-algebra ideal I, A is Hausdorff with respect to F.

SEMISIMPLE, ARCHIMEDEAN, AND SEMILOCAL PSEUDO MV-ALGEBRAS

The concepts of semisimple, Archimedean, and semilocal pseudo MV-algebras are investigated and many interesting facts concerning them are given. Pseudo MV -algebras were introduced by G. Georgescu and A. Iorgulescu in (6) and in- dependently by J. Rachunek in (8) (there they are called generalized MV -algebras or, for short, GMV -algebras) as a non-commutative generalization of MV -algebras. This work was intended as an attempt to order some notions appearing in the theory of these al- gebras. Semisimple pseudo MV -algebras and Archimedean pseudo MV -algebras are ex- amples of such notions. In Section 3 we give some characterizations of semisimple pseudo MV -algebras. Archimedean pseudo MV -algebras are investigated and characterized in Sec- tion 4. It is shown that in the case of pseudo MV -algebras the notion of Archimedean is equivalent with the notion of Archimedean in the Belluce sense, that occurs in the theory of MV -algebras, and both are equivalent with the notion of semisimple. Section 5 is devoted to introduce and characterize semilocal pseudo MV -algebras, the concept generalizing a simi- lar one from the theory of MV -algebras. For the convenience of the reader, in Section 2 we give the relevant material needed in the sequel, thus making our exposition self-contained.

On the semisimple tensor product of MV-algebras

In this note we show how the MV-algebraic tensor product can be used to define natural adjunctions between the algebraic structures of Łukasiewicz logic with product. In order to do this, we define the semisimple tensor PMV-algebra of a semisimple MV-algebra.

Duality Theory and Skeleta for Semisimple MV-Algebras

We start from Marra–Spada duality between semisimple MV-algebras and Tychonoff spaces, and we consider the particular cases when the $$\omega $$ -skeleta of the MV-algebras are restricted in some way. In particular we consider antiskeletal MV-algebras, that is, the ones whose $$\omega $$ -skeleton is trivial.

Affine representations of $${\ell}$$ ℓ -groups and MV-algebras

Affine representations for archimedean \({\ell}\)-groups and semisimple MV-algebras via embedding theorems are presented; they are simple to work with but powerful enough to express significant properties of our studied objects. Indeed, we focus on the space of particular homomorphisms between an archimedean \({\ell}\)-group (a semisimple MV-algebra, respectively) and a vector lattice (a Riesz MV-algebra, respectively), i.e., the set of the generalized states, providing a general framework.

An extension of Stone Duality to fuzzy topologies and MV-algebras

In this paper we introduce the concept of MV-topology, a special class of fuzzy topological spaces, and prove a proper extension of Stone Duality to the categories of limit cut complete MV-algebras and Stone MV-spaces, namely, zero-dimensional compact Hausdorff MV-topological spaces. Then we describe the category of limit cut complete MV-algebras, and show that it is a completion—namely, a reflective subcategory with faithful reflection—of the category of semisimple MV-algebra. Last, we compose our duality with other known ones, thus obtaining new categorical equivalences and dualities involving categories of MV-algebras.

Scalar extensions for algebraic structures of Łukasiewicz logic

In this paper we study the tensor product for MV-algebras, the algebraic structures of Łukasiewicz ∞-valued logic. Our main results are: the proof that the tensor product is preserved by the categorical equivalence between the MV-algebras and abelian lattice-order groups with strong unit and the proof of the scalar extension property for semisimple MV-algebras. We explore consequences of these results for various classes of MV-algebras and lattice-ordered groups enriched with a product operation.

The Riesz Hull of a Semisimple MV-Algebra

Abstract MV-algebras and Riesz MV-algebras are categorically equivalent to abelian lattice-ordered groups with strong unit and, respectively, with Riesz spaces (vector-lattices) with strong unit. A standard construction in the literature of lattice-ordered groups is the vector-lattice hull of an archimedean latticeordered group. Following a similar approach, in this paper we define the Riesz hull of a semisimple MV-algebra.

The Chinese Remainder Theorem for Strongly Semisimple MV-Algebras and Lattice-Groups

Abstract An MV-algebra (equivalently, a lattice-ordered Abelian group with a distinguished order unit) is strongly semisimple if all of its quotients modulo finitely generated congruences are semisimple. All MV-algebras satisfy a Chinese Remainder Theorem, as was first shown by Keimel four decades ago in the context of lattice-groups. In this note we prove that the Chinese Remainder Theorem admits a considerable strengthening for strongly semisimple structures.

On tense MV-algebras

The main aim of this article is to study tense MV-algebras which are just MV-algebras with new unary operations G and H which express a universal time quantifiers. Tense MV-algebras were introduced by D. Diaconescu and G. Georgescu. Using a new notion of an fm-function between MV-algebras we settle a half of their Open problem about representation for some classes of tense MV-algebras, i.e., we show that any tense semisimple MV-algebra is induced by a time frame analogously to classical works in this field of logic. As a by-product we obtain a new characterization of extremal states on MV-algebras. Our method gives a general framework for representing functions with the so-called Jauch–Piron property (including MV-morphisms) between MV-algebras.

Bouligand–Severi k-tangents and strongly semisimple MV-algebras

An algebra A is said to be strongly semisimple if every principal congruence of A is an intersection of maximal congruences. We give a geometrical characterisation of strongly semisimple MV-algebras in terms of Bouligand–Severi k-tangents. The latter are a k-dimensional generalisation of the classical Bouligand–Severi tangents.

Operators on MV-algebras and their representations

A crucial problem concerning (tense) operators on MV-algebras is their representations. Having an MV-algebra with (tense) operators, we can ask if there exists a frame yielding a representation of this MV-algebra. We solve this problem for semisimple MV-algebras.

The Dual Adjunction between MV-algebras and Tychonoff Spaces

We offer a proof of the duality theorem for finitely presented MV-algebras and rational polyhedra, a folklore and yet fundamental result. Our approach develops first a general dual adjunction between MV-algebras and subspaces of Tychonoff cubes, endowed with the transformations that are definable in the language of MV-algebras. We then show that this dual adjunction restricts to a duality between semisimple MV-algebras and closed subspaces of Tychonoff cubes. The duality theorem for finitely presented objects is obtained by a further specialisation. Our treatment is aimed at showing exactly which parts of the basic theory of MV-algebras are needed in order to establish these results, with an eye towards future generalisations.

Normal and complete Boolean ambiguity algebras and MV-pairs

In 2006, both Gejza Jenca and Thomas Vetterlein, building on different hypotheses, represented MV-algebras through the quotient of a Boolean algebra B by a subgroup of the group of all automorphisms of B. It is shown in this article that Vetterlein's constructions are particular cases of Jenca's and that they give semisimple MV-algebras.

Core of Coalition Games on MV-algebras

Coalition games are generalized to semisimple MV-algebras. Coalitions are viewed as [0, 1]-valued functions on a set of players, which enables to express a degree of membership of a player in a coalition. Every game is a real-valued mapping on a semisimple MV-algebra. The goal is to recover the so-called core: a set of final distributions of payoffs, which are represented by measures on the MV-algebra. A class of sublinear games are shown to have a non-empty core and the core is completely characterized in certain special cases. The interpretation of games on propositional formulas in Łukasiewicz logic is introduced.

MV -Algebras of Continuous Functions and l-Monoids

Abstract. A. Di Nola & S.Sessa [8] showed that two compact spaces X and Y arehomeomorphic iff the MV -algebras C(X,I) and C(Y,I) of continuous functions defined onX and Y respectively are isomorphic. And they proved that A is a semisimple MV -algebraiff A is a subalgebra of C(X) for some compact Hausdorff space X. In this paper, firstlyby use of functorial argument, we show these characterization theorems. Furthermorewe obtain some other functorial results between topological spaces and MV -algebras.Secondly as a classical problem, we find a necessary and sufficient condition on a givenresiduated l-monoid that it is segmenently embedded into an l-group with order unit. 1. IntroductionAn MV-algebra is a universal algebra (A,+,·,∗,0,1) of (2,2,1,0,0) type suchthat (A,+,0) is an abelian monoid and moreover, x+1 = 1,x ∗∗ = x,0 ∗ = 1,x+x ∗ =1,x·y = (x ∗ +y ∗ ) and x+x ∗ y = y+y ∗ x for all x,y ∈ A. By setting x∨y = x+x ∗ yand x∧y = x(x ∗ +y) we have (A,∨,∧,0,1) as a bounded distributive lattice.The system of MV-algebras is a kind of better system in the sense that closedunder subalgebras, quotients and products and the free MV-algebra with a denu-merable set of generators can be described by MV-algebras of continuous I = [0,1]-valued functions on the Hilbert cube [11]. Furthermore, the variety of MV-algebrasis a Malcev variety and has the congruence regularity [10].In the first part of this paper, we establish a dual-adjunction (η,e) : S ‘ C fromthe subcategory X of Tychonoff spaces of Top into the subcategory A of semi-simple MV-algebras of M

Every state on semisimple MV-algebra is integral

Integral representation theorem will be established for finitely additive probability measures (states) on semisimple MV-algebras. This result generalizes the well-known theorem of Butnariu and Klement in case of σ -order continuous states on tribes of fuzzy sets. Precisely, it will be demonstrated that every state on a separating clan of continuous fuzzy sets arises as an integral with respect to a unique Borel probability measure. The key technique leading to this result exploits the geometrical–topological properties of the state space: the set of all states on every MV-algebra forms a Bauer simplex.

SPECTRAL DUALITIES OF MV-ALGEBRAS

Hong and Nel in [8] obtained a number of spectral dualities between a cartesian closed topological category X and a category of algebras of suitable type in X in accordance with the original formalism of Porst and Wischnewsky[12]. In this paper, there arises a dual adjointness S <TEX>$\vdash$</TEX> C between the category X = Lim of limit spaces and that A of MV-algebras in X. We firstly show that the spectral duality: <TEX>$S(A)^{op}{\simeq}C(X^{op})$</TEX> holds for the dualizing object K = I = [0,1] or K = 2 = {0, 1}. Secondly, we study a duality between the category of Tychonoff spaces and the category of semi-simple MV-algebras. Furthermore, it is shown that for any <TEX>$X\;\in\;Lim\;(X\;{\neq}\;{\emptyset})\;C(X,\;I)$</TEX> is densely embedded into a cube <TEX>$I^/H/$</TEX>, where H is a set.