Vectorial MV-algebras

 In this paper we introduce the concepts of vectorial MV-algebra, vectorial complete MV-algebra, modulus of continuity and continuous function with values in a vectorial complete MV-algebra. The main result of this paper is a Weierstrass type approximation theorem for functions taking values on vectorial complete MV-algebras. As examples, we construct such a structure on the RGB model (this represents the colours model of a pixel on the screen) and on the space of fuzzy subsets of a set. 2000 Mathematics Subject Classification 06D35; 06B23; 41A10 Current address: Armatei Romane 5, 3700 Oradea, Romania e-mail address: dnoje@uoradea.ro     

Extensions of MV-algebras

 We introduce MV-extensions of MV-algebras and their ideals as a special kind of extensions in the category of partial abelian monoids and show their relations with the extensions of the corresponding ℓ-groups. Dedicated to Prof. Ján Jakubík on the occasion of his 80th birthday Supported by grant VEGA 2/3163/23.     

Roughness in MV-algebras

In this paper, by considering the notion of an MV-algebra, we consider a relationship between rough sets and MV-algebra theory. We introduce the notion of rough ideal with respect to an ideal of an MV-algebra, which is an extended notion of ideal in an MV-algebra, and we give some properties of the lower and the upper approximations in an MV-algebra.     

On pseudo MV-algebras

 We study pseudo MV-algebras introduced recently by Georgescu and Iorgulescu as a non-commutative generalization of MV-algebras. We introduce a partial binary operation which model the addition in pseudo MV-algebras. In the paper, we give basic properties of such an addition. We find conditions which entail the commutativity of pseudo MV-algebras, i.e., when they are classical MV-algebras. We study ideals and the conditions when a pseudo MV-algebra is representable. Finally, we introduce states and show how they are connected with the normal ideals.     

Boolean dominated MV-algebras

In this paper we study an obvious generalization of the hyperarchimedean MV-algebras: boolean dominated MV-algebras. Particularly we point out the wide difference between the class of the hyperarchimedean MV-algebras and the class of the boolean dominated MV-algebras.     

Local pseudo MV-algebras

 The pseudo MV-algebras were defined in [12] as non-commutative extensions of MV-algebras. In this paper we define the local pseudo MV-algebras and we suggest a classification for these structures. The subclass of perfect pseudo MV-algebras is deeply investigated. Extending the Di Nola–Lettieri result for MV-algebras [8], we prove that the category of l-groups is equivalent to a subcategory of perfect pseudo MV-algebras.     

Measures on MV-algebras

 We study sequentially continuous measures on semisimple M V-algebras. Let A be a semisimple M V-algebra and let I be the interval [0,1] carrying the usual Łukasiewicz M V-algebra structure and the natural sequential convergence. Each separating set H of M V-algebra homomorphisms of A into I induces on A an initial sequential convergence. Semisimple M V-algebras carrying an initial sequential convergence induced by a separating set of M V-algebra homomorphisms into I are called I-sequential and, together with sequentially continuous M V-algebra homomorphisms, they form a category SM(I). We describe its epireflective subcategory ASM(I) consisting of absolutely sequentially closed objects and we prove that the epireflection sends A into its distinguished σ-completion σ _H (A). The epireflection is the maximal object in SM(I) which contains A as a dense subobject and over which all sequentially continuous measures can be continuously extended. We discuss some properties of σ _H (A) depending on the choice of H. We show that the coproducts in the category of D-posets [9] of suitable families of I-sequential M V-algebras yield a natural model of probability spaces having a quantum nature. The motivation comes from probability: H plays the role of elementary events, the embedding of A into σ _H (A) generalizes the embedding of a field of events A into the generated σ-field σ(A), and it can be viewed as a fuzzyfication of the corresponding results for Boolean algebras in [8, 11, 14]. Sequentially continuous homomorphisms are dual to generalized measurable maps between the underlying sets of suitable bold algebras [13] and, unlike in the Loomis–Sikorski Theorem, objects in ASM(I) correspond to the generated tribes (no quotient is needed, no information about the elementary events is lost). Finally, D-poset coproducts lift fuzzy events, random functions and probability measures to events, random functions and probability measures of a quantum nature. Supported by VEGA Grant 2/7193/01     

Retract mappings of projectable MV-algebras

For MV-algebras (algebras of multivalued Lukasiewicz logics) we apply the same terminology and notation as in [3] and [8]. Retracts and retract mappings of abelian lattice ordered groups were studied in [4], cf. also [6], [7]; for the case of multilattice groups and cyclically ordered groups cf. [1] and [5]. To each MV-algebra ? there corresponds an abelian lattice ordered group G with a strong unit u such that (under the notation as in [8]), ? = ?₀(G,u) (cf. also Section 1 below). In [2], a different (but equivalent) system of axioms for defining the notion of MV-algebra was applied; instead of ?₀(G,u), the notation Γ(G,u) was used. In the present paper we investigate the relations between retract mappings of a projectable MV-algebra ? and the retract mappings of the corresponding lattice ordered group G.     

Tensor products of MV-algebras

 The structure of MV-module over a PMV-algebra was defined in [5]. In this paper we test this notion in the important problem of scalar extension. The tensor product ⊗ _mv defined in [11] is generalized and another tensor product, ⊗ _o , is studied in relation with the corresponding tensor product of ℓ-groups [9]. Using tensor products we get the free MV-modules over a totally ordered PMV-algebra. Dedicated to Prof. Ján Jakubíik on the occasion of his 80th birthday     

A Representation Theorem for MV-algebras

An MV-pair is a pair (B,G) where B is a Boolean algebra and G is a subgroup of the automorphism group of B satisfying certain conditions. Let ~ _G be the equivalence relation on B naturally associated with G. We prove that for every MV-pair (B,G), the effect algebra B/ ~ _G is an MV-effect algebra. Moreover, for every MV-effect algebra M there is an MV-pair (B,G) such that M is isomorphic to B/ ~ _G . This research is supported by grant VEGA G-1/3025/06 of MŠ SR, Slovakia and by the Science and Technology Assistance Agency under the contract No. APVT-51-032002.     

Representation of IF-probability on MV-algebras

A probability theory on IF-events has been constructed by Grzegorzewski and Mrówka in [2] and axiomatically characterized by B. Riečan in [6]. More general situation was studied by B. Riečan in [5]. He introduced the IF-probability, which is decomposed onto two probabilities on Lukasiewicz tribe. The aim of this paper is to define the IF-probability and IF-observable on MV-algebras. We show that some known results of [7] can be applied also for these IF-events.     

软集合和软MV-代数

通过将Molodtsov在1999年提出的软集合理论应用于MV-代数理论,提出了软MV-代数和软MV-子代数的概念,并且得到了相关的一些性质。     

Ring-like structures corresponding to pseudo MV-algebras

The natural bijective correspondence between MV-algebras and pseudorings is generalized to the case of pseudo MV-algebras. We first introduce the concepts of non-commutative pseudorings and strong pseudo De Morgan algebras. Then, we show that they both correspond to pseudo MV-algebras via symmetric difference in a natural bijective way. This work was supported by NSFC Major Research Program 60496324; NSFC Key Research Program 60736011; NSFC 60603002; Pre-973 Project 2001CCA03000; 863 High-Tech Project 2001AA113130; 973 Project 2001CB312004; CAS Brain and Mind Science Project.     

Conditional independence in probability theory on MV-algebras

Conditional independence relation of observables on an MV-algebra with product is formulated and basic properties are proven. An analogy with a classical definition of conditional independence of random variables is discussed and the MV-algebraic concept of conditioning is further explored.     

A note on observables on MV-algebras

 A criterion when a subset of an MV-algebra is contained in the range of a (finitely additive) observable is given. It is used to prove that every at most countable subset of any MV-algebra is contained in the range of an observable. A much stronger result holds for any bold fuzzy algebra [0,1] ^S , which is whole contained in the range of a (σ-additive) observable.     

Hahn–Banach theorems for MV-algebras

We introduce the MV-seminorms and we prove Hahn–Banach type extension results for linear functions and MV-algebra homomorphisms dominated by MV-seminorms.     

Continuous approximations of product implication in MV-algebras with product

Recently, MV-algebras with product have been investigated from different points of view. In particular, in [EGM01], a variety resulting from the combination of MV-algebras and product algebras (see [H98]) has been introduced. The elements of this variety are called ŁΠ-algebras. Even though the language of ŁΠ-algebras is strong enough to describe the main properties of product and of ukasiewicz connectives on [0,1], the discontinuity of product implication introduces some problems in the applications, because a small error in the data may cause a relevant error in the output. In this paper we try to overcome this difficulty, substituting the product implication by a continuous approximation of it. We investigate the resulting algebras, called ŁΠ_q -algebras, and we show that these algebras constitute a quasivariety which is generated by the class of all ŁΠ _q -algebras whose lattice reduct is the unit interval [0,1] with the usual order.     

Almost everywhere convergence in MV-algebras with product

 First, we examine in general MV-algebras with product the concept of the lower and upper limits known from the theory of tribes of fuzzy sets. Then we compare two concepts of almost everywhere convergence and we show that they coincide.