Interval valued (\in,\in\!\vee\,q) -fuzzy filters of pseudo BL-algebras

We introduce the concept of quasi-coincidence of a fuzzy interval value with an interval valued fuzzy set. By using this new idea, we introduce the notions of interval valued -fuzzy filters of pseudo BL-algebras and investigate some of their related properties. Some characterization theorems of these generalized interval valued fuzzy filters are derived. The relationship among these generalized interval valued fuzzy filters of pseudo BL-algebras is considered. Finally, we consider the concept of implication-based interval valued fuzzy implicative filters of pseudo BL-algebras, in particular, the implication operators in Lukasiewicz system of continuous-valued logic are discussed.     

Extremal states on bounded residuated \ell-monoids with general comparability

Bounded residuated lattice ordered monoids (RlR\ell-monoids) are a common generalization of pseudo-BLBL-algebras and Heyting algebras, i.e. algebras of the non-commutative basic fuzzy logic (and consequently of the basic fuzzy logic, the Łukasiewicz logic and the non-commutative Łukasiewicz logic) and the intuitionistic logic, respectively. We investigate bounded RlR\ell-monoids satisfying the general comparability condition in connection with their states (analogues of probability measures). It is shown that if an extremal state on Boolean elements fulfils a simple condition, then it can be uniquely extended to an extremal state on the RlR\ell-monoid, and that if every extremal state satisfies this condition, then the RlR\ell-monoid is a pseudo-BLBL-algebra.     

The axiomatic characterizations on L-generalized fuzzy neighborhood system-based approximation operators

The rough sets based on L-fuzzy relations and L-fuzzy coverings are the two most well-known L-fuzzy rough sets. Quite recently, we prove that some of these rough sets can be unified into one framework—rough sets based on L-generalized fuzzy neighborhood systems. So, the study on the rough sets based on L-generalized fuzzy neighborhood system has more general significance. Axiomatic characterization is the foundation of L-fuzzy rough set theory: the axiom sets of approximation operators guarantee the existence of L-fuzzy relations, L-fuzzy coverings that reproduce the approximation operators. In this paper, we shall give an axiomatic study on L-generalized fuzzy neighborhood system-based approximation operators. In particular, we will seek the axiomatic sets to characterize the approximation operators generated by serial, reflexive, unary and transitive L-generalized fuzzy neighborhood systems, respectively.     

A Generalization of Local Fuzzy Structures

The class of bounded residuated lattice ordered monoids Rl-monoids) contains as proper subclasses the class of pseudo BL-algebras (and consequently those of pseudo MV-algebras, BL-algebras and MV-algebras) and of Heyting algebras. In the paper we introduce and investigate local bounded Rl-monoids which generalize local algebras from the above mentioned classes of fuzzy structures. Moreover, we study and characterize perfect bounded Rl-monoids.     

Fuzzy -ary polygroups related to fuzzy points

Recently, fuzzy n-ary sub-polygroups were introduced and studied by Davvaz, Corsini and Leoreanu-Fotea [B. Davvaz, P. Corsini, V. Leoreanu-Fotea, Fuzzy n-ary sub-polygroups, Comput. Math. Appl. 57 (2008) 141–152]. Now, in this paper, the concept of (,q)-fuzzy n-ary sub-polygroups, -fuzzy n-ary sub-polygroups and fuzzy n-ary sub-polygroup with thresholds of an n-ary polygroup are introduced and some characterizations are described. Also, we give the definition of implication-based fuzzy n-ary sub-polygroups in an n-ary polygroup, in particular, the implication operators in Łukasiewicz system of continuous-valued logic are discussed.     

The law of large numbers for fuzzy processes and the estimation problem

We study how the law of large numbers can be extended to the case of fuzzy numbers. We work with L-fuzzy sets, where L is a CL_∞ commutative monoid with an operation 1. We define the 1-independence of fuzzy numbers. We find, when 1 is the operation Λ of the lattice, that the arithmetical mean of Λ-independent fuzzy numbers does not converge. We think that this case could be a limit one; we show, for particular cases of the 1-operation, that generally convergence holds. We build estimators for fuzzy models; we particularly consider the “minimum fuzziness estimator.” We study the convergence of these estimators and compare them by way of the fuzzy subsets they induce. We lastly study the case where we have dependent fuzzy numbers and particularly the case of linearly filtered “1-autoindependent” fuzzy processes.     

The three-dimensional fuzzy sets and their cut sets

In this paper, a new kind of L-fuzzy set is introduced which is called the three-dimensional fuzzy set. We first put forward four kinds of cut sets on the three-dimensional fuzzy sets which are defined by the 4-valued fuzzy sets. Then, the definitions of 4-valued order nested sets and 4-valued inverse order nested sets are given. Based on them, the decomposition theorems and representation theorems are obtained. Furthermore, the left interval-valued intuitionistic fuzzy sets and the right interval-valued intuitionistic fuzzy sets are introduced. We show that the lattices constructed by these two special L-fuzzy sets are not equivalent to sublattices of lattice constructed by the interval-valued intuitionistic fuzzy sets. Finally, we show that the three-dimensional fuzzy set is equivalent to the left interval-valued intuitionistic fuzzy set or the right interval-valued intuitionistic fuzzy set.     

Finiteness based results in BL-algebras

BL-algebras were introduced by P. Hájek as algebraic structures of Basic Logic. The aim of this paper is to survey known results about the structure of finite BL-algebras and natural dualities for varieties of BL-algebras. Extending the notion of ordinal sum of BL-algebras , we characterize a class of finite BL-algebras, actually BL-comets, which can be seen as a generalization of finite BL-chains. Then, just using BL-comets, we can represent any finite BL-algebra A as a direct product of BL-comets. This result can be seen as a generalization of the representation of finite MV-algebras as a direct product of MV-chains. Then we consider the varieties generated by one finite non-trivial totally ordered BL-algebra. For each of these varieties, we show the existence of a strong duality. As an application of the dualities, the injective and the weak injective members of these classes are described.     

Characterizations of regular ordered semigroups in terms of (α, β)-fuzzy generalized bi-ideals

In this paper, the concept of an (α, β)-fuzzy generalized bi-ideal in an ordered semigroup is introduced, which is a generalization of the concept of a fuzzy generalized bi-ideal in an ordered semigroup. Using this concept, some characterization theorems are provided. The upper/lower parts of an (∈, ∈ ∨ q)-fuzzy generalized bi-ideal are introduced and some characterizations of regular ordered semigroups are given. Also, we consider the concept of implication-based fuzzy generalized bi-ideals in an ordered semigroup. In particular, the implication operators in Lukasiewicz system of continuous-valued logic are discussed.     

Set functors,L-fuzzy set categories,and generalized terms

In this work, we generalize previous constructions of fuzzy set categories, introduced in [1], by considering L-fuzzy sets in which the values of the characteristic functions run on a completely distributive lattice, rather than in the unit real interval. Later, these L-fuzzy sets are used to define the L-fuzzy categories, which are proven to be rational. In the final part of the paper, the L-fuzzy functors given by the extension principles are provided with a structure of monad which is used, together with the functorial definition of the term monad, to provide monad compositions as a basis for a notion of generalised terms.     

Interval-valued fuzzy n-ary subhypergroups of n-ary hypergroups

This paper provides a continuation of ideas presented by Davvaz and Corsini (J Intell Fuzzy Syst 18(4):377–382, 2007). Our aim in this paper is to introduce the concept of quasicoincidence of a fuzzy interval value with an interval-valued fuzzy set. This concept is a generalized concept of quasicoincidence of a fuzzy point within a fuzzy set. By using this new idea, we consider the interval-valued (∈, ∈ ∨q)-fuzzy n-ary subhypergroup of a n-ary hypergroup. This newly defined interval-valued (∈, ∈ ∨q)-fuzzy n-ary subhypergroup is a generalization of the usual fuzzy n-ary subhypergroup. Finally, we consider the concept of implication-based interval-valued fuzzy n-ary subhypergroup in an n-ary hypergroup; in particular, the implication operators in ￡ukasiewicz system of continuous-valued logic are discussed.     

On the strict logic foundation of fuzzy reasoning

This paper focuses on the logic foundation of fuzzy reasoning. At first, a new complete first-order fuzzy predicate calculus system K^* corresponding to the formal system L^* is built. Based on the many-sort system K_ms^* corresponding to K^*, the triple I methods of FMP and FMT for fuzzy reasoning and their consistency are formalized, thus fuzzy reasoning is put completely and rigorously into the logic framework of fuzzy logic.The author is indebted to anonymous referee for his useful comments which have helped to improve the paper.     

Observations on non-commutative fuzzy logic

The paper presents some results on the logic psBL (pseudo-basic fuzzy logic, the generalization of BL not assuming commutativity of conjunction) and on the analogous logic psMTL – a non-commutative version of the monoidal t-norm logic MTL of Esteva and Godo.Partial support of the project No LN00A056 (ITI) of the Ministry of Education (MMT) of the Czech Republic is acknowledged. Thanks are due to F. Esteva and the anonymous referee for their comments on a draft of this paper.     

L-fuzzy numbers and their properties

In this paper, the notions of L-fuzzy convex sets and L-fuzzy numbers are introduced where L is a completely distributive lattice. The notions of [0, 1]-fuzzy convex sets and [0, 1]-fuzzy numbers are generalized. Furthermore their properties and characterizations are presented in terms of cut sets of L-fuzzy sets.     

Intuitionistic (<Emphasis Type="Italic">S</Emphasis>, <Emphasis Type="Italic">T</Emphasis>)-fuzzy hyperquasigroups

The concept of intuitionistic fuzzy subhyperquasigroups in a hyperquasigroup with respect to an s-norm and a t-norm on intuitionistic fuzzy sets is introduced and their properties of such hyperquasigroups are studied. Intuitionistic (S, T)-fuzzy relations on a hyperquasigroup G are discussed. In particular, we investigate connections hyperquasigroups with binary quasigroups.     

Fuzzy roughness of n-ary hypergroups based on a complete residuated lattice

This paper considers the relationships among L-fuzzy sets, rough sets and n-ary hypergroup theory. Based on a complete residuated lattice, the concept of (invertible) L-fuzzy n-ary subhypergroups of a commutative n-ary hypergroup is introduced and some related properties are presented. The notions of lower and upper L-fuzzy rough approximation operators with respect to an L-fuzzy n-ary subhypergroup are introduced and studied. Then, a new algebraic structure called (invertible) L-fuzzy rough n-ary subhypergroups is defined, and the (strong) homomorphism of lower and upper L-fuzzy rough approximation operators is studied.     

Fuzzy n-ary hypergroups related to fuzzy points

In this paper, we introduce and study a new sort of fuzzy n-ary sub-hypergroups of an n-ary hypergroup, called $(\in,\in \vee q)In this paper, we introduce and study a new sort of fuzzy n-ary sub-hypergroups of an n-ary hypergroup, called ( ? , ? úq)(\in,\in \vee q)-fuzzy n-ary sub-hypergroup. By using this new idea, we consider the ( ? , ? úq)(\in,\in\vee q)-fuzzy n-ary sub-hypergroup of a n-ary hypergroup. This newly defined ( ? , ? úq)(\in,\in \vee q)-fuzzy n-ary sub-hypergroup is a generalization of the usual fuzzy n-ary sub-hypergroup. Finally, we consider the concept of implication-based fuzzy n-ary sub-hypergroup in an n-ary hypergroup and discuss the relations between them, in particular, the implication operators in £\poundsukasiewicz system of continuous-valued logic are discussed.     

A new view of fuzzy hypernear-rings

In this survey article, we first introduce the concept of quasi-coincidence of a fuzzy interval value with an interval valued fuzzy set. This concept is a generalized concept of quasi-coincidence of a fuzzy point within a fuzzy set. By using this new idea, we consider the interval valued (∈,∈∨q)-fuzzy sub-hypernear-rings (hyperideals) of a hypernear-ring, and hence, a generalization of a fuzzy sub-near-ring (ideal) is given. Some related properties of fuzzy hypernear-rings are described. Finally, we consider the concept of implication-based interval valued fuzzy sub-hypernear-rings (hyperideals) in a hypernear-ring, in particular, the implication operators in Lukasiewicz system of continuous-valued logic are discussed.