Local algebras in the representation of MV-algebras

In this paper we present several results about local MV-algebras, extending existing results given for MV-chains. The role of local MV-algebras in sheaf representation and weak boolean product is stressed and the relationship of local MV-algebras with varieties of MV-algebras is analyzed. Presented by S. Pulmannova. Received November 11, 2005; accepted in final form December 20, 2005.     

Weakly linear quantum MV-algebras

Quantum MV-algebras (QMV-algebras) are a non lattice-theoretic generalization of MV-algebras (multi-valued algebras) and a non-idempotent generalization of orthomodular lattices. In this paper we construct a finite basis for the variety generated by the class of all weakly linear quantum MV-algebras.Dedicated to the memory of Wim BlokReceived October 12, 2000; accepted in final form October 3, 2004.This revised version was published online in August 2005 with a corrected cover date.     

MV-algebras: a variety for magnitudes with archimedean units

Chang’s MV-algebras, on the one hand, are the algebras of the infinite-valued Łukasiewicz calculus and, on the other hand, are categorically equivalent to abelian lattice-ordered groups with a distinguished strong unit, for short, unital ℓ-groups. The latter are a modern mathematization of the time-honored euclidean magnitudes with an archimedean unit. While for magnitudes the unit is no less important than the zero element, its archimedean property is not even definable in first-order logic. This gives added interest to the equivalent representation of unital ℓ-groups via the equational class of MV-algebras. In this paper we survey several applications of this equivalence, and various properties of the variety of MV-algebras.Dedicated to the Memory of Wim BlokReceived August 26, 2003; accepted in final form October 3, 2004.This revised version was published online in August 2005 with a corrected cover date.     

Dedekind Complete Commutative BCK-algebras

We study Dedekind complete commutative BCK-algebras with the relative cancellation property and their connection with corresponding universal groups. We shall characterize Dedekind orthogonally complete atomic and Archimedean BCK-algebras, generalizing results of Jakubík known for MV-algebras. Finally, we characterize those Dedekind complete and atomic commutative BCK-algebras that are isomorphic to direct products of basic BCK-chains, generalizing a result of Cignoli for MV-algebras.     

Join-semilattices whose sections are residuated PO-monoids

We generalize the concept of an integral residuated lattice to join-semilattices with an upper bound where every principal order-filter (section) is a residuated semilattice; such a structure is called a sectionally residuated semilattice. Natural examples come from propositional logic. For instance, implication algebras (also known as Tarski algebras), which are the algebraic models of the implication fragment of the classical logic, are sectionally residuated semilattices such that every section is even a Boolean algebra. A similar situation rises in case of the Lukasiewicz multiple-valued logic where sections are bounded commutative BCK-algebras, hence MV-algebras. Likewise, every integral residuated (semi)lattice is sectionally residuated in a natural way. We show that sectionally residuated semilattices can be axiomatized as algebras (A, r, →, ⇝, 1) of type 〈3, 2, 2, 0〉 where (A, →, ⇝, 1) is a {→, ⇝, 1}-subreduct of an integral residuated lattice. We prove that every sectionally residuated lattice can be isomorphically embedded into a residuated lattice in which the ternary operation r is given by r(x, y, z) = (x · y) ∨ z. Finally, we describe mutual connections between involutive sectionally residuated semilattices and certain biresiduation algebras. This work was supported by the Czech Government via the project MSM6198959214.     

关于格蕴涵代数与BCK-代数

证明了格蕴涵代数与有界可换 Ｂ Ｃ Ｋ代数是两类相互等价的代数系统,借此得到了一类 Ｂ Ｃ Ｋ代数的结构定理     

The finite embeddability property for residuated lattices, pocrims and BCK-algebras

A class of algebras has the finite embeddability property (FEP) if every finite partial subalgebra of an algebra in the class can be embedded into a finite algebra in the class. We investigate the relationship of the FEP with the finite model property (FMP) and strong finite model property (SFMP).? For quasivarieties the FEP and the SFMP are equivalent, and for quasivarieties with equationally definable principal relative congruences the three notions FEP, FMP and SFMP are equivalent. The variety of intuitionistic linear algebras –which is known to have the FMP–fails to have the FEP, and hence the SFMP as well. The variety of integral intuitionistic linear algebras (also known as the variety of residuated lattices) does possess the FEP, and hence also the SFMP. Similarly contrasting statements hold for various subreduct classes. In particular, the quasivarieties of pocrims and of BCK-algebras possess the FEP. As a consequence, the universal theories of the classes of residuated lattices, pocrims and BCK-algebras are decidable. Received February 16, 2001; accepted in final form November 2, 2001. RID="h1" ID="h1"The second author was supported by a postdoctoral research fellowship of the National Research Foundation of South Africa, hosted by the University of Illinois at Chicago.     

Generalizations of Boolean products for lattice-ordered algebras

It is shown that the Boolean center of complemented elements in a bounded integral residuated lattice characterizes direct decompositions. Generalizing both Boolean products and poset sums of residuated lattices, the concepts of poset product, Priestley product and Esakia product of algebras are defined and used to prove decomposition theorems for various ordered algebras. In particular, we show that FL_w-algebras decompose as a poset product over any finite set of join irreducible strongly central elements, and that bounded n-potent GBL-algebras are represented as Esakia products of simple n-potent MV-algebras.     

On the structure of generalized BL-algebras

A generalized BL - algebra (or GBL-algebra for short) is a residuated lattice that satisfies the identities . It is shown that all finite GBL-algebras are commutative, hence they can be constructed by iterating ordinal sums and direct products of Wajsberg hoops. We also observe that the idempotents in a GBL-algebra form a subalgebra of elements that commute with all other elements. Subsequently we construct subdirectly irreducible noncommutative integral GBL-algebras that are not ordinal sums of generalized MV-algebras. We also give equational bases for the varieties generated by such algebras. The construction provides a new way of order-embedding the lattice of -group varieties into the lattice of varieties of integral GBLalgebras. The results of this paper also apply to pseudo-BL algebras. This paper is dedicated to Walter Taylor. Received March 7, 2005; accepted in final form July 25, 2005.     

Complete Commutative Basic Algebras

By a basic algebra is meant an MV-like algebra (A, ⊕, ¬, 0) of type 〈2, 1, 0〉 derived in a natural way from bounded lattices having antitone involutions on their principal filters. In the previous paper (Botur and Hala?, Mult. Valued Log. Soft Comp., 2007) we have shown that finite basic algebras for which the operation ⊕ is commutative are MV-algebras. In this paper we generalize this result by considering commutative basic algebras for which the underlying lattice is complete.     

A condition for constructing formally real monoid algebras

In this paper we demonstrate that every positive totally ordered commutative monoid on 2 generators satisfying a weak cancellation property is a convex Rees quotient of a sub-monoid of a totally ordered Abelian group. In [1], the current author, along with Evans, Konikoff, Mathis, and Madden, employed the work of Hion, [5], to demonstrate that the monoid ring of all finite formal sums over a totally ordered domain is a formally real totally ordered ring providing the totally ordered monoid satisfies this weak cancellation property and is a convex Rees quotient of a sub-monoid of a totally ordered Abelian group. Therefore, we provide here significant information about a condition for the construction of formally real totally ordered monoid algebras. Received November 4, 2003; accepted in final form November 18, 2004.     

Notes on “A survey of fuzzy implication algebras and their axiomatization”

In this note, we present a correction of Theorem 2 in the paper [1], and show that all the conditions (I1)–(I5) in the definition of fuzzy implication algebras are independent of each other. In addition, we prove that the class of all commutative fuzzy implication algebras forms an equational algebra class.     

Representation theory of MV-algebras

In this paper we develop a general representation theory for MV-algebras. We furnish the appropriate categorical background to study this problem. Our guide line is the theory of classifying topoi of coherent extensions of universal algebra theories. Our main result corresponds, in the case of MV-algebras and MV-chains, to the representation of commutative rings with unit as rings of global sections of sheaves of local rings. We prove that any MV-algebra is isomorphic to the MV-algebra of all global sections of a sheaf of MV-chains on a compact topological space. This result is intimately related to McNaughton’s theorem, and we explain why our representation theorem can be viewed as a vast generalization of McNaughton’s theorem. In spite of the language used in this abstract, we have written this paper in the hope that it can be read by experts in MV-algebras but not in sheaf theory, and conversely.     

Many-valued quantum algebras

We deal with algebras of the same signature as MV-algebras which are a common extension of MV-algebras and orthomodular lattices, in the sense that (i) A bears a natural lattice structure, (ii) the elements a for which is a complement in the lattice form an orthomodular sublattice, and (iii) subalgebras whose elements commute are MV-algebras. We also discuss the connections with lattice-ordered effect algebras and prove that they form a variety. Supported by the Research and Development Council of the Czech Government via the project MSM6198959214.     

Generalized orderings and rings of fractions

Every total ordering of a commutative domain can be extended uniquely to its field of fractions. This result is extended in two directions. Firstly, the notion of a total ordering is generalized so that a nonzero element can have more than two signs (in fact, these signs form a group). Secondly, commutative domains are replaced by noncommutative ones and we consider the following types of rings of fractions: Ore extensions, maximal (right or two-sided) rings of fractions, division hulls of free algebras and epic fields. Throughout the paper several examples are given to illustrate the theory. Received January 8, 2005; accepted in final form November 1, 2005.     

Canonical extensions of double quasioperator algebras: An algebraic perspective on duality for certain algebras with binary operations

The context for this paper is a class of distributive lattice expansions, called double quasioperator algebras (DQAs). The distinctive feature of these algebras is that their operations preserve or reverse both join and meet in each coordinate. Algebras of this type provide algebraic semantics for certain non-classical propositional logics. In particular, MV-algebras, which model the ?ukasiewicz infinite-valued logic, are DQAs.Varieties of DQAs are here studied through their canonical extensions. A variety of this type having additional operations of arity at least 2 may fail to be canonical; it is already known, for example, that the variety of MV-algebras is not. Non-canonicity occurs when basic operations have two distinct canonical extensions and both are necessary to capture the structure of the original algebra. This obstruction to canonicity is different in nature from that customarily found in other settings. A generalized notion of canonicity is introduced which is shown to circumvent the problem. In addition, generalized canonicity allows one to capture on the canonical extensions of DQAs the algebraic operations in such a way that the laws that these obey may be translated into first-order conditions on suitable frames. This correspondence may be seen as the algebraic component of duality, in a way which is made precise.In many cases of interest, binary residuated operations are present. An operation h which, coordinatewise, preserves ∨ and 0 lifts to an operation which is residuated, even when h is not. If h also preserves binary meet then the upper adjoints behave in a functional way on the frames.     

Topological duality for Tarski algebras

In this paper we will generalize the representation theory developed for finite Tarski algebras given in [7]. We will introduce the notion of Tarski space as a generalization of the notion of dense Tarski set, and we will prove that the category of Tarski algebras with semi-homomorphisms is dually equivalent to the category of Tarski spaces with certain closed relations, called T-relations. By these results we will obtain that the algebraic category of Tarski algebras is dually equivalent to the category of Tarski spaces with certain partial functions. We will apply these results to give a topological characterization of the subalgebras. Received August 21, 2005; accepted in final form December 5, 2006.     

A characterization of free generating sets in MV-algebras

MV-algebras are a generalization of Boolean algebras. As is well known, a free generating set for a Boolean algebra is characterized by the following simple algebraic condition: whenever A and B are finite disjoint subsets of X then . Our aim in this note is to give a similar characterization of free generating sets in MV-algebras. Received January 30, 2005; accepted in final form March 13, 2007.     

States on commutative basic algebras

The paper deals with states on commutative basic algebras that are a non-associative generalization of MV-algebras or, in other words, the algebraic semantics for a fuzzy logic which generalizes the ?ukasiewicz logic in that the conjunction is not associative. States are defined in the same way as Mundici's states on MV-algebras as normalized finitely additive [0,1]-valued functions, and some results analogous to the results that are known from MV-algebras are proved.     

正定关联BCI—代数

本文是作者［１］和［２］的继续，引入了正定关联ＢＣＩ－代数的概念，并证明了：正定关联ＧＢＣＫ－代数类和Ｐ－半单ＢＣＩ－代数类是正定关联ＢＣＩ－代数类的真子类。