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.     

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.     

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.     

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.     

Ideals of an EMV-semiring

ABSTRACT

In this paper, congruences, ideals, and prime ideals of an EMV-semiring and of its associated EMV-algebra are studied. Then EMV-semirings are characterized and it is proved that each EMV-semiring can be embedded into a direct product of a family of MV-semirings as an EMV-semiring. Moreover, another representation of EMV-semirings are presented by EMV-semirings of continuous sections in a sheaf of commutative semirings whose stalks are localizations of EMV-semirings over prime ideals. Also, using the categorical equivalence between EMV-semirings and EMV-algebras, a representation of EMV-algebras are obtained.     

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.     

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.     

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.     

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.     

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.     

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.     

Basic fuzzy logic and <Emphasis Type="Italic">BL</Emphasis>-algebras II

 Three new (easy) results about the computational complexity of basic propositional fuzzy logic BL are presented. An important formula of predicate logic is shown 1-true in all interpretations over saturated BL-chains but is not a BL-1-tautology, i.e. is not 1-true in a safe interpretation over a non-saturated BL-algebra. Partial support of the grant No. A1030004/00 of the Grant Agency of the Academy of Science of the Czech Republic is acknowledged.     

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.     

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.     

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.     

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.     

Poincaré–Birkhoff–Witt theorem for Leibniz -algebras

We construct the universal enveloping algebra of a Leibniz n-algebra and we prove that the category of modules over this algebra is equivalent to the category of representations.We also give a proof of the Poincaré–Birkhoff–Witt theorem for universal enveloping algebras of finite-dimensional Leibniz n-algebras using Gröbner bases in a free associative algebra.