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.     

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.     

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.     

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.     

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.     

Semirings and pseudo MV algebras

In this paper, we describe the relationships between pseudo MV algebras and semirings. We also give definitions of automata on lattice ordered semirings, prove that the family of K-Languages is closed under union, and discuss the conditions for the closedness of families of K-languages under intersection, generalized intersection and reversal operations.     

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.     

Compactness in <Emphasis Type="Italic">MV</Emphasis>-algebras and in their non-commutative generalizations

 In the paper we study MV-algebras and their non-commutative generalizations, GMV-algebras, in which every element is compact. Further, we characterize finite MV-algebras as archimedean GMV-algebras which are 0-meet compact. Dedicated to Prof. Ján Jakubík on the occasion of his 80th birthday Supported by the Council of Czech Government, J 14/98: 15100011.     

On two versions of the Loomis–Sikorski Theorem for algebraic structures

We present two versions of the Loomis–Sikorski Theorem, one for monotone σ-complete generalized pseudo effect algebras with strong unit satisfying a kind of the Riesz decomposition property. The second one is for Dedekind σ-complete positive pseudo Vitali spaces with strong unit. For any case we can find an appropriate system of nonnegative bounded functions forming an algebra of the given type with the operations defined by points that maps epimorphically onto the algebra. The paper has been supported by the Center of Excellence SAS—Physics of Information—I/2/2005, the grant VEGA No. 2/6088/26 SAV, the Slovak Research and Development Agency under the contract No. APVV-0071-06, Slovak-Italian Project No. 15:“Algebraic and logical systems of soft computing”, and MURST, project “Analisi Reale”.     

State operators on GMV algebras

Flaminio and Montagna recently introduced state MVMV algebras as MVMV algebras with an internal state in the form of a unary operation. Di Nola and Dvurečenskij further presented a stronger variation of state MVMV algebras called state-morphism MVMV algebras. In the paper we present state GMVGMV algebras and state-morphism GMVGMV algebras which are non-commutative generalizations of the mentioned algebras.     

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.     

Torsion classes of <Emphasis Type="Italic">MV</Emphasis>-algebras

 Torsion classes of MV-algebras are defined as radical classes which are closed with respect to homomorphisms; in this paper we investigate their relations to radical classes of lattice ordered groups and to varieties of MV-algebras. Supported by Grant VEGA 1/9056/02.     

Minimal prime state filters and state radicals on state pseudo $${ }$$-algebras

The aim of this paper is the study of some classes of state filters of a state pseudo BL-algebra. The concepts of minimal prime state filter and of state hyperarchimedean pseudo BL-algebra are introduced and a characterization of a state hyperarchimedean pseudo BL-algebra is presented. Also, we define the notion of a state radical of a state filter of a state pseudo BL-algebra, we present a characterization of a state radical and some of its properties. The algebra of state radicals of a state pseudo BL-algebra is studied.     

Fuzzy prime ideals of pseudo-<Emphasis Type="Italic">MV</Emphasis> algebras

The aim of this paper is to introduce the notion of fuzzy prime ideals of pseudo-MV algebras and investigate some of its properties.      

TL-filters of integral residuated l-monoids

In this paper, we introduce and discuss the concept of TL-filters of integral residuated l-monoids. First we study some basic properties of TL-filters and give the formula for calculating the TL-filters generated by L-subsets. Then we discuss some properties of TL-filters under homomorphisms and study the relationship between TL-filters and T-congruence L-relations on integral commutative residuated l-monoids.     

Every Linear Pseudo BL-Algebra Admits a State

We show that every linear pseudo BL-algebra, hence every representable one, admits a state and is good. This solves positively the problem on the existence of states raised in Dvurečenskij and Rachůnek (Probabilistic averaging in bounded communitative residuated ℓ-monoids, 2006), and gives a partial answer to the problem on good pseudo BL-algebras from [Di Nola, Georgescu and Iorgulescu (Multiple Val Logic 8:715–750, 2002) Problem 3.21]. Moreover, we present that every saturated linear pseudo BL-algebra can be expressed as an ordinal sum of Hájek’s type of irreducible pseudo linear pseudo BL-algebras. The paper has been supported by the Center of Excellence SAS—Physics of Information—I/2/2005, the grant VEGA no. 2/3163/23 SAV and by Science and Technology Assistance Agency under the contract no. APVT-51-032002. Bratislava, Slovakia.     

Pseudo d-algebras

We introduce the notion of a pseudo d-algebra as a generalization of the idea of a d-algebra. The class of pseudo d-algebras can be included in the class of coupled d-algebras. We show that the class of pseudo d-algebras behaves well with respect to standard mappings such as direct products and standard mappings such as homomorphisms. Indeed, we are able to prove a fundamental theorem of homomorphisms and we are able to identify kernel objects as d-ideals in this theory. We also discuss several large classes of examples of pseudo d-algebras which occur in different settings and we identify properties sufficient for or equivalent to membership in important subclasses of the class of pseudo d-algebras in these settings.