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.     

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.     

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.     

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.     

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.     

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.     

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.     

Measures on MV-algebras

 We study sequentially continuous measures on semisimple M V-algebras. Let A be a semisimple M V-algebra and let I be the interval [0,1] carrying the usual Łukasiewicz M V-algebra structure and the natural sequential convergence. Each separating set H of M V-algebra homomorphisms of A into I induces on A an initial sequential convergence. Semisimple M V-algebras carrying an initial sequential convergence induced by a separating set of M V-algebra homomorphisms into I are called I-sequential and, together with sequentially continuous M V-algebra homomorphisms, they form a category SM(I). We describe its epireflective subcategory ASM(I) consisting of absolutely sequentially closed objects and we prove that the epireflection sends A into its distinguished σ-completion σ _H (A). The epireflection is the maximal object in SM(I) which contains A as a dense subobject and over which all sequentially continuous measures can be continuously extended. We discuss some properties of σ _H (A) depending on the choice of H. We show that the coproducts in the category of D-posets [9] of suitable families of I-sequential M V-algebras yield a natural model of probability spaces having a quantum nature. The motivation comes from probability: H plays the role of elementary events, the embedding of A into σ _H (A) generalizes the embedding of a field of events A into the generated σ-field σ(A), and it can be viewed as a fuzzyfication of the corresponding results for Boolean algebras in [8, 11, 14]. Sequentially continuous homomorphisms are dual to generalized measurable maps between the underlying sets of suitable bold algebras [13] and, unlike in the Loomis–Sikorski Theorem, objects in ASM(I) correspond to the generated tribes (no quotient is needed, no information about the elementary events is lost). Finally, D-poset coproducts lift fuzzy events, random functions and probability measures to events, random functions and probability measures of a quantum nature. Supported by VEGA Grant 2/7193/01     

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 routing schemes allow broadcasting with linear message-complexity

Summary. The purpose of compact routing is to provide a labeling of the nodes of a network and a way to encode the routing tables, so that routing can be performed efficiently (e.g., on shortest paths) whilst keeping the memory-space required to store the routing tables as small as possible. In this paper, we answer a long-standing conjecture by showing that compact routing may also assist in the performance of distributed computations. In particular, we show that a network supporting a shortest path interval routing scheme allows broadcasting with a message-complexity of O(n), where n is the number of nodes of the network. As a consequence, we prove that O(n) messages suffice to solve leader-election for any graph labeled by a shortest path interval routing scheme, improving the previous known bound of O(m+n). A general consequence of our result is that a shortest path interval routing scheme contains ample structural information to avoid developing ad-hoc or network-specific solutions for basic problems that distributed systems must handle repeatedly. Received: December 2000 / Accepted: July 2001     

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.     

D-homomorphisms and atomic σ-complete boolean D-posets

We study observables as σ-D-homomorphisms defined on Boolean D-posets of subsets into a Boolean D-poset. We show that given an atomic σ-complete Boolean D-poset ? with the countable set of atoms there exist a σ-complete Boolean D-poset of subsets ? and a σ-D-homomorphism h from ? onto ?, more precisely we can choose ? = ?(?), which gives an analogy of the Loomis–Sikorski representation theorem for Boolean σ-algebras. We show also that any atomic σ-complete Boolean D-poset with the countable set of atoms is the range of a σ-homomorphism defined on a σ-complete Boolean D-poset of fuzzy sets which gives another type of the Loomis–Sikorski theorem.     

Fuzzy n-fold ideals in BCH-algebras

In this paper, we introduce the notion of fuzzy n-folds (P, implicative and fantastic) ideals in BCH-algebras which is a natural generalization of notion of n-folds (P, implicative and fantastic) ideals in BCH-algebras and we stated and proved some theorems which determines the relationship between these notions. Finally we give some computational Algorithms for these notions.     

<Emphasis Type="Italic">Smart CAPs for Smart Its</Emphasis> – Context Detection for Mobile Users

Context detection for mobile users plays a major role for enabling novel, human-centric interfaces. For this, we introduce a context detection scheme applicable in a self-organized sensor network, which is formed of disseminated, computer empowered sensors, referred to as Smart-Its [1]. Context-detection takes place without requiring any central point of control, and supports push as well as pull modes. Our solution is based on an in-network composition approach relying on so-called smart Context-Aware Packets (sCAPs). These packets act as a uniform interchange format, and allow single sensors to share sensed data and to cooperate to build up a meaningful context model from manifold inputs. sCAPs travel through the sensor network governed by an enclosed retrieving plan, specifying which sensors to visit in order to gain a specific piece of context information. For enhanced flexibility, the retrieving plan itself may be dynamically altered in accordance with past sensor readings.     

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.     

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.