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.     

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.     

Very and more or less in non-commutative fuzzy logic

In this paper we consider fuzzy subsets of a universe as L-fuzzy subsets instead of [ 0, 1 ]-valued, where L is a complete lattice. We enrich the lattice L by adding some suitable operations to make it into a pseudo-BL algebra. Since BL algebras are main frameworks of fuzzy logic, we propose to consider the non-commutative BL-algebras which are more natural for modeling the fuzzy notions. Based on reasoning with in non-commutative fuzzy logic we model the linguistic modifiers such as very and more or less and give an appropriate membership function for each one by taking into account the context of the given fuzzy notion by means of resemblance L-fuzzy relations.     

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.     

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.     

The Variety Generated by Perfect BL-Algebras: an Algebraic Approach in a Fuzzy Logic Setting

BL-algebras are the Lindenbaum algebras of the propositional calculus coming from the continuous triangular norms and their residua in the real unit interval. Any BL-algebra is a subdirect product of local (linear) BL-algebras. A local BL-algebra is either locally finite (and hence an MV-algebra) or perfect or peculiar. Here we study extensively perfect BL-algebras characterizing, with a finite scheme of equations, the generated variety. We first establish some results for general BL-algebras, afterwards the variety is studied in detail. All the results are parallel to those ones already existing in the theory of perfect MV-algebras, but these results must be reformulated and reproved in a different way, because the axioms of BL-algebras are obviously weaker than those for MV-algebras.     

n-Fold obstinate filters in BL-algebras

In this paper, we introduced the notion of n-fold obstinate filter in BL-algebras and we stated and proved some theorems, which determine the relationship between this notion and other types of n-fold filters in a BL-algebra. We proved that if F is a 1-fold obstinate filter, then A/F is a Boolean algebra. Several characterizations of n-fold fantastic filters are given, and we show that A is a n-fold fantastic BL-algebra if A is a MV-algebra (n ≥ 1) and A is a 1-fold positive implicative BL-algebra if A is a Boolean algebra. Finally, we construct some algorithms for studying the structure of the finite BL-algebras and n-fold filters in finite BL-algebras.     

Basic fuzzy logic and BL-algebras

 The many-valued propositional logic BL (basic fuzzy logic) is investigated. It is known to be complete for tautologies over BL-algebras (particular residuated lattices). Each continuous t-norm on [0,1] determines a BL-algebra; such algebras are called t-algebras. Two additional axioms B1, B2 are found such that BL+(B1,B2) is complete for tautologies over t-algebras. It remains open whether B1, B2 are provable in BL.     

Metrizability on (semi)topological BL-algebras

In this paper we define the notion of quasifilter neighborhoods on (semi)topological BL-algebras and state and prove some of their properties. Finally, using the concept of quasifilter, we find some conditions under which a BL-algebra will become metrizable.     

Separation axioms in (semi)topological quotient BL-algebras

In this paper, we study the relationship between separation axioms and (semi)topological quotient BL-algebras. We bring some conditions under which a (semi)topological quotient BL-algebra becomes a T ₁-space or Hausdorff or regular or normal. Also, we use maximum condition to get a Hausdorff or regular or normal (semi)topological quotien BL-algebra.     

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.     

Factor congruences in BCK-algebras

In this paper, we characterize factor congruences in the quasivariety of BCK-algebras. As an application we prove that the free algebra over an infinite set of generators is indecomposable in any subvariety of BCK-algebras. We also study the decomposability of free algebras in the variety of hoop residuation algebras and its subvarieties. We prove that free algebras in a non k-potent subvariety of are indecomposable while finitely generated free algebras in k-potent subvarieties have a unique non-trivial decomposition into a direct product of two factors, and one of them is the two-element implication algebra. This paper is partially supported by Universidad Nacional del Sur and CONICET.     

Generalized fuzzy filters of <Emphasis Type="Italic">R</Emphasis><Subscript>0</Subscript>-algebras

In this paper, we introduce the notions of interval valued -fuzzy filters and interval valued -fuzzy Boolean (implicative) filters in R ₀-algebras and investigate some of their related properties. Some characterization theorems of these generalized fuzzy filters are derived. In particular, we prove that an interval valued fuzzy set F in R ₀-algebras is an interval valued -fuzzy Boolean filter if and only if it is an interval valued -fuzzy implicative filter.     

On very true operators on pocrims

Hájek introduced the logic enriching the logic BL by a unary connective vt which is a formalization of Zadeh’s fuzzy truth value “very true”. algebras, i.e., BL-algebras with unary operations, called vt-operators, which are among others subdiagonal, are an algebraic counterpart of Partially ordered commutative integral residuated monoids (pocrims) are common generalizations of both BL-algebras and Heyting algebras. The aim of our paper is to introduce and study algebraic properties of pocrims endowed by “very-true” and “very-false”-like operators. Research is supported by the Research and Development Council of Czech Government via project MSN 6198959214.     

States on semi-divisible residuated lattices

Given a residuated lattice L, we prove that the subset MV(L) of complement elements x ^* of L generates an MV-algebra if, and only if L is semi-divisible. Riečan states on a semi-divisible residuated lattice L, and Riečan states on MV(L) are essentially the very same thing. The same holds for Bosbach states as far as L is divisible. There are semi-divisible residuated lattices that do not have Bosbach states. These results were obtained when the authors visited Academy of Science, Czech Republic, Institute of Comp. Sciences in Autumn 2006.     

WBR0-代数的Pt-模和Ps-模表示

三角模和蕴涵算子是逻辑代数中的一对相互关联的基本算子,而WBR0-代数是仅以蕴涵算子为基本算子定义于一般集合上的逻辑代数。对WBR0-代数进行了再研究。给出了WBR0-代数的Pt-模（偏序集上的t-模）的表现形式;在偏序集上通过 Pt-模结合蕴涵算子给出了WBR0-代数的表示形式;证明了WBR0-代数原始定义中的二元运算[⊕]是WBR0-代数的Ps-模（偏序集上的s-模）。结果对进一步探讨WBR0-代数及其相关的逻辑代数有积极的参考价值。     

Functions represented by BL-algebra formulas: characterization and approximate representation

The paper is a contribution to the study of functional systems of BL-algebras on [0,1] extended by constants from [0,1] and suprema and infima over (infinite) sets of different instances of the same formula. We prove that the class of functions which are represented by thus specified formulas coincides with the class of extensional functions. Moreover, the class of these functions coincides with the class of Lipschitz continuous functions in the case of BL-algebra whose t-norm has a continuous additive generator. We also consider an approximate representation of functions, represented by BL-formulas with a possibly infinite length, by discrete normal forms and suggest an algorithmic approach to its construction. This paper has been partially supported by grant IAA1187301 of the GA AV ČR.     

(Weak) Dual hyper <Emphasis Type="Italic">K</Emphasis>-ideals

In this note by considering the notion of (weak) dual hyper K-ideal, we obtain some related results. After that we determine the relationships between (weak) dual hyper K-ideals and (weak) hyper K-ideals. Finally, we give a characterization of hyper K-algebras of order 3 or 4 based on the (weak) dual hyper K-ideals.     

BL-代数的n-重模糊蕴涵滤子

引入BL-代数的n-重模糊蕴涵滤子概念,讨论了n-重模糊蕴涵滤子的性质,给出若干等价刻画,给出了验证n-重模糊蕴涵滤子的一个简单条件。用n-重模糊蕴涵滤子对n-重蕴涵BL-代数进行了刻画,证明了一个模糊滤子是一个n-重模糊蕴涵滤子当且仅当对应的商代数是一个n-重蕴涵BL-代数。