Complete MV-algebras Research Articles

MV-frames

Abstract Each complete MV-algebra is naturally equipped with a structure of a frame. For this reason, we shall call such MV-algebras MV-frames and investigate some of their main frame properties. We completely characterize algebraic MV-frames as well as regular MV-frames. In addition, we consider nuclei on MV-frames in general and on MV-frames of ideals of Łukasiewicz rings in particular. Finally, we use the Chang–Mundici functor to explore the frame structures of complete unital lattice-ordered groups. 2010 Mathematics Subject Classification. 06D35, 06E15, 06D50

MV-algebras and their corresponding Bézout domains

The paper studies the interplay between MV-algebras, Bézout domains, and Abelian l -groups, which was initiated by Y. C. Yang in 2006. For each MV-algebra M = Γ ( G , u ) , we delve into some properties of a subset of a Bézout domain corresponding to M. We study Boolean elements and ideals within the MV-algebra, focusing on their correspondence to subsets of Bézout domains. This investigation aids us in classifying Bézout domains about directly indecomposable MV-algebras or subdirect products of MV-algebras. We establish that each MV-algebra associated with a Noetherian Bézout domain is finite. In conclusion, from a ring-theoretical perspective, we present characteristics of perfect MV-algebras, ( H , 1 ) -perfect MV-algebras, hyperarchimedean MV-algebras, and complete MV-algebras.

Intermediate quantifiers and valid syllogisms on EQ-algebras

Intermediate quantifiers are expressions of natural language, for example “most, almost all, many, a few” using which we quantify a number of some objects in a given universe. We have shown in [23] that all valid syllogisms with intermediate quantifiers are a consequence of only two algebraic inequalities and one equality. The result was obtained in the formalism of Lukasiewicz fuzzy type theory whose truth values form a linearly ordered complete MV-algebra. In this paper we will prove that the same holds if we replace MV-algebra by a much more general IEQ-algebra (involutive EQ-algebra).

Rough Semiring-Valued Fuzzy Sets with Application

Many of the new fuzzy structures with complete MV-algebras as value sets, such as hesitant, intuitionistic, neutrosophic, or fuzzy soft sets, can be transformed into one type of fuzzy set with values in special complete algebras, called AMV-algebras. The category of complete AMV-algebras is isomorphic to the category of special pairs (R,R∗) of complete commutative semirings and the corresponding fuzzy sets are called (R,R∗)-fuzzy sets. We use this theory to define (R,R∗)-fuzzy relations, lower and upper approximations of (R,R∗)-fuzzy sets by (R,R∗)-relations, and rough (R,R∗)-fuzzy sets, and we show that these notions can be universally applied to any fuzzy type structure that is transformable to (R,R∗)-fuzzy sets. As an example, we also show how this general theory can be used to determine the upper and lower approximations of a color segment corresponding to a particular color.

A category of complete residuated lattice-value neighborhood groups

In this paper, considering L a complete residuated lattice, we present a lattice-valued category TNG (resp., TTNG) of (topological) ⊤-neighborhood groups, where the object is defined as a group equipped with a (topological) ⊤-neighborhood space such that the group operations are continuous with respect to the (topological) ⊤-neighborhood space. It is proved that: (1) The ⊤-neighborhood space associated with a ⊤-neighborhood group is topological, so the category TNG is equivalent to the category TTNG. Hence TTNG is redundant, and we need only discuss TNG. (2) The category TNG has nice characterizations, localization and uniformization. (3) The category TNG has initial structure, so it is a topological category, and each initial structure has an ordered representation. (4) The category NG of neighborhood groups can be embedded in TNG as a reflective subcategory. (5) The category TNG can be embedded in the category SLNG of stratified L-neighborhood groups as a reflective subcategory when the underlying lattice L is a meet-continuous lattice. (6) The category TNG is equivalent to the category StrLTOPG of strong L-topological groups when L is a complete MV-algebra.

Compactness in MV-topologies: Tychonoff theorem and Stone–Čech compactification

In this paper, we discuss some questions about compactness in MV-topological spaces. More precisely, we first present a Tychonoff theorem for such a class of fuzzy topological spaces and some consequence of this result, among which, for example, the existence of products in the category of Stone MV-spaces and, consequently, of coproducts in the one of limit cut complete MV-algebras. Then we show that our Tychonoff theorem is equivalent, in ZF, to the Axiom of Choice, classical Tychonoff theorem, and Lowen's analogous result for lattice-valued fuzzy topology. Last, we show an extension of the Stone-Cech compactification functor to the category of MV-topological spaces, and we discuss its relationship with previous works on compactification for fuzzy topological spaces.

Stone MV-algebras and strongly complete MV-algebras

Characterizations of compact Hausdorff topological MV-algebras, Stone MV-algebras, and MV-algebras that are isomorphic to their profinite completions are established. It is proved that compact Hausdorff topological MV-algebras are products (both topological and algebraic) of copies [0, 1] with the interval topology and finite Łukasiewicz chains with the discrete topology. Going one step further, we also prove that Stone MV-algebras are products (both topological and algebraic) of finite Łukasiewicz chains with the discrete topology. Finally, it is proved that an MV-algebra is isomorphic to its profinite completion if and only if it is profinite and each of its maximal ideals of finite rank is principal.

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.

On topological MV-algebras and topological ℓ-groups

We prove a representation theorem for complete MV-algebras endowed with a Hausdorff order continuous (o.c.) locally convex topology which admits a 0-neighbourhood base consisting of sublattices and deduce from this a representation theorem for Hausdorff o.c. locally solid Dedekind complete ℓ-groups which admits a 0-neighbourhood base consisting of sublattices.

Fuzzy logic models in a category of fuzzy relations

We investigate interpretations \({\|\psi\|_{\mathcal E}}\) of formulas ψ in a first order fuzzy logic in models \({\mathcal {E}}\) which are based on objects of a category SetR(Ω) which consists of Ω-sets, i.e. sets with similarity relations with values in a complete MV-algebra Ω and with morphisms defined as special fuzzy relations between Ω-sets. The interpretations \({\|\psi\|_\mathcal {E}}\) are then morphisms in a category SetR(Ω) from some Ω-set to the object \({(\Omega,\leftrightarrow)}\). We define homomorphisms between models in a category SetR(Ω) and we prove that if \({\varphi : \mathcal {E}_1\rightarrow \mathcal {E}_2}\) is a (special) homomorphism of models in a category SetR(Ω) then there is a relation between interpretations \({\|\psi\|_{\mathcal {E}_i}}\) of a formula ψ in models \({\mathcal {E}_i}\).

Complete Subobjects of Fuzzy Sets Over MV-Algebras

A subobjects structure of the category Ω-FSet of Ω-fuzzy sets over a complete MV-algebra $$\Omega = \left( {L,\Lambda , \vee , \otimes , \to } \right)$$ is investigated, where an Ω-fuzzy set is a pair A = (A, δ) such that A is a set and δ: A × A → Ω is a special map. Special subobjects (called complete) of an Ω-fuzzy set A which can be identified with some characteristic morphisms A → Ω* = (L × L, μ) are then investigated. It is proved that some truth-valued morphisms $$_\Omega :\Omega ^ * \to \Omega ^ * , \cap _\Omega , \cup _\Omega :\Omega ^ * \times \Omega ^ * \to \Omega ^ * $$ are characteristic morphisms of complete subobjects.

Some properties of fuzzy net-convergences

In this paper, we introduce the fuzzy limit degrees in L-fuzzy topologies using complete MV-algebras. We investigate the properties of net-convergences (the degrees of fuzzy convergent, fuzzy cluster, fuzzy adherent points and fuzzy limits).

A Cantor-Bernstein Theorem for σ-Complete MV-Algebras

The Cantor-Bernstein theorem was extended to σ-complete boolean algebras by Sikorski and Tarski. Chang's MV-algebras are a nontrivial generalization of boolean algebras: they stand to the infinite-valued calculus of Lukasiewicz as boolean algebras stand to the classical two-valued calculus. In this paper we further generalize the Cantor-Bernstein theorem to σ-complete MV-algebras, and compare it to a related result proved by Jakubik for certain complete MV-algebras.

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.

The degrees of fuzzy net-convergences on complete MV-algebras

In this paper, we introduce the degrees of fuzzy net-convergences in L-fuzzy topologies using complete MV-algebras. We investigate the relationships among the degrees of fuzzy convergent, fuzzy cluster and fuzzy adherent points. We study the properties of net-convergences.

MV-ALGEBRA VALUED FILTER THEORY

Abstract Let M be a complete MV-algebra with square roots. Then every M-valued filter can be represented as “intersection” of an appropriate family of maximal M-valued filters. In the case of M = [0,1] maximal [0,1]-valued filters and finitely additive probability measures come to the same thing.

Semi-simple and complete MV-algebras

Semi-simple and complete MV-algebras