Subreducts of MV-algebras with product and product residuation

Recently, MV-algebras with product have been investigated from different points of view. In particular, in [EGM01], a variety resulting from the combination of MV-algebras and product algebras (see [H98]) has been introduced. The elements of this variety are called ŁΠ-algebras. In this paper we treat subreducts of ŁΠ-algebras, with emphasis on quasivarieties of subreducts whose basic operations are continuous in the order topology. We give axiomatizations of the most interesting classes of subreducts, and we connect them with other algebraic classes of algebras, like f-rings and Wajsberg hoops, as well as to structures of co-infinitesimals of ŁΠ-algebras. In some cases, connections are given by means of equivalences of categories.Dedicated to the Memory of Wim BlokReceived June 19, 2002; accepted in final form November 29, 2004.This revised version was published online in August 2005 with a corrected cover date.     

Localization of MV-algebras and lu-groups

The aim of the present paper is to define the localization MV-algebra of an MV-algebra A with respect to a topology on A; also, following the categorical equivalence between the category of lu-groups and the category of MV-algebras, we define the analogous notion for lu-groups. In Section 5 we prove that the maximal MV-algebra of quotients (defined in [7]) and the MV-algebra of fractions relative to an -closed system (defined in [6]) are MV-algebras of localization.In the last part of this paper (Section 6) we prove analogous results for lu-groups.     

On Product MV-Algebras

In this paper we apply the notion of the product MV-algebra in accordance with the definition given by B. Riean. We investigate the convex embeddability of an MV-algebra into a product MV-algebra. We found sufficient conditions under which any two direct product decompositions of a product MV-algebra have isomorphic refinements.     

Projective limits of MV-spaces

An MV-space is a topological space X such that there exists an MV-algebra A whose prime spectrum Spec A is homeomorphic to X. The characterization of the MV-spaces is an important open problem.We shall prove that any projective limit of MV-spaces in the category of spectral spaces is an MV-space. In this way, we obtain new classes of MV-spaces related to some preservation properties of the Belluce functor.     

Decidable and undecidableMV-algebras

In (2) Chang introduced the notion ofMV-algebra in order to show completeness of many-valued Lukasiewicz-logic. Using a characterisation ofMV-algebras in a simpler language, proposed by H. Läuchli, this article shows the undecidability of the elementary theory ofMV-algebras by reducing it to the undecidability of lattice-ordered abelian groups (Gurevich). Further it is shown that the wordproblem forMV-algebras is solvable and that the elementary theories of finite-orderMV-algebras are decidable.I wish to thank Prof. H. Läuchli for his idea to describeMV-aigebras by the truncated difference and for a lot of fruitful talks about the subject.Presented by S. Burris.     

Monadic <Emphasis Type="Italic">GMV</Emphasis>-algebras

Monadic MV-algebras are an algebraic model of the predicate calculus of the Łukasiewicz infinite valued logic in which only a single individual variable occurs. GMV-algebras are a non-commutative generalization of MV-algebras and are an algebraic counterpart of the non-commutative Łukasiewicz infinite valued logic. We introduce monadic GMV-algebras and describe their connections to certain couples of GMV-algebras and to left adjoint mappings of canonical embeddings of GMV-algebras. Furthermore, functional MGMV-algebras are studied and polyadic GMV-algebras are introduced and discussed. The first author was supported by the Council of Czech Government, MSM 6198959214.     

Perfect residuated lattice ordered monoids

Bounded Rℓ-monoids form a large subclass of the class of residuated lattices which contains certain of algebras of fuzzy and intuitionistic logics, such as GMV-algebras (= pseudo-MV-algebras), pseudo-BL-algebras and Heyting algebras. Moreover, GMV-algebras and pseudo-BL-algebras can be recognized as special kinds of pseudo-MV-effect algebras and pseudo-weak MV-effect algebras, i.e., as algebras of some quantum logics. In the paper, bipartite, local and perfect Rℓ-monoids are investigated and it is shown that every good perfect Rℓ-monoid has a state (= an analogue of probability measure).     

Monadic Bounded Commutative Residuated ℓ-monoids

Bounded commutative residuated ℓ-monoids are a generalization of algebras of propositional logics such as BL-algebras, i.e. algebraic counterparts of the basic fuzzy logic (and hence consequently MV-algebras, i.e. algebras of the Łukasiewicz infinite valued logic) and Heyting algebras, i.e. algebras of the intuitionistic logic. Monadic MV-algebras are an algebraic model of the predicate calculus of the Łukasiewicz infinite valued logic in which only a single individual variable occurs. We introduce and study monadic residuated ℓ-monoids as a generalization of monadic MV-algebras. Jiří Rachůnek was supported by the Council of Czech Goverment MSM 6198959214.     

Weak homogeneity and Pierce’s theorem for MV-algebras

In this paper we prove a theorem on weak homogeneity of MV-algebras which generalizes a known result on weak homogeneity of Boolean algebras. Further, we consider a homogeneity condition for MV-algebras which is defined by means of an increasing cardinal property.     

Prime spectra of non-commutative generalizations of MV-algebras

Non-commutative generalizations of MV-algebras were introduced by G. Georgescu and A. Iorgulesco as well as by the author; the generalizations are equivalent and are called GMV-algebras. We show that GMV-algebras can be considered as special cases of Grishin algebras. As MV-algebras are algebraic models of the Łukasiewicz logic and Grishin algebras have the analogous role for the classical bilinear logic, GMV-algebras correspond to a non-commutative logic between the above logics. Further, by A. Dvurečenskij, any GMV-algebra is isomorphic to an interval of an l-group, which in general is not commutative. This generalizes D. Mundici's representation of MV-algebras by means of intervals of abelian l-groups. In the paper (using this representation) we describe the properties of prime ideal spectra of GMV-algebras and of their factor algebras and ideals and prove that the spectrum of closed ideals of any GMV-algebra is homeomorphic to that of a completely distributive GMV-algebra. Received January 4, 2001; accepted in final form May 2, 2002.     

MV-Test Spaces Versus MV-Algebras

In analogy with effect algebras, we introduce the test spaces and MV-test spaces. A test corresponds to a hypothesis on the propositional system, or, equivalently, to a partition of unity. We show that there is a close correspondence between MV-algebras and MV-test spaces.     

Banaschewski’s theorem for <Emphasis Type="Italic">GMV</Emphasis>-algebras

Banaschewski’s theorem concerns subdirect product decompositions of lattice ordered groups. In the present paper we deal with the analogous investigation for the case of generalized MV -algebras (GMV -algebras, in short); we apply this notion in the sense studied by Galatos and Tsinakis.     

Direct summands and retract mappings of generalized <Emphasis Type="Italic">MV</Emphasis>-algebras

In the present paper we deal with generalized MV-algebras (GMV-algebras, in short) in the sense of Galatos and Tsinakis. According to a result of the mentioned authors, GMV-algebras can be obtained by a truncation construction from lattice ordered groups. We investigate direct summands and retract mappings of GMV-algebras. The relations between GMV-algebras and lattice ordered groups are essential for this investigation. Supported by VEGA Agency grant 1/2002/05. This work has been partially supported by the Slovak Academy of Sciences via the project Center of Excellence-Physics of Information, grant I/2/2005.     

State-Homomorphisms on MV-Algebras

Riecan [12] and Chovanec [1] investigated states in MV-algebras. Earlier, Riecan [11] had dealt with analogous ideas in D-posets. In the monograph of Riecan and Neubrunn [13] (Chapter 9) the notion of state is applied in the theory of probability on MV-algebras. We remark that a different definition of a state in an MV-algebra has been applied by Mundici [9], [10] (namely, the condition (iii) from Definition 1.1 above was not included in his definition of a state; in other words, only finite additivity was assumed). Below we work with the definition from [13]; but, in order to avoid terminological problems we use the term "state-homomorphism" (instead of "state"). The author is indebted to the referee for his suggestion concerning terminology. Let be an MV-algebra which is defined on a set A with card A>1. In the present paper we show that there exists a one-to-one correspondence between the system of all state-homomorphisms on and the system of all -closed maximal ideals of . For MV-algebras we apply the notation and the definitions as in Gluschankof [3]. The relations between MV-algebras and abelian lattice ordered groups (cf. Mundici [8]) are substantially used in the present paper.     

On Free MV-Algebras

In the present paper we show that free MV-algebras can be constructed by applying free abelian lattice ordered groups.     

Isometries of generalized <Emphasis Type="Italic">MV</Emphasis>-algebras

In this paper we investigate the relations between isometries and direct product decompositions of generalized MV-algebras.     

On Varieties of Pseudo MV-Algebras

In this paper we investigate the relation between the lattice of varieties of pseudo MV-algebras and the lattice of varieties of lattice ordered groups.     

Complemented congruences on Ockham algebras

An Ockham algebra that satisfies the identity is called a K_{n, m}-algebra. Generalizing some results obtained in [2], J. Varlet and T. Blyth, in [3, Chapter 8], study congruences on K_{1, 1}-algebras. In particular, they describe the complement (when it exists) of a principal congruence and characterize these congruences that are complemented. In this paper we study the same question for K_{n, m}-algebras. Received March 24, 2005; accepted in final form April 28, 2005.     

Equational type characterization for σ-complete MV-algebras

In the framework of algebras with infinitary operations, an equational base for the category of σ-complete MV-algebras is given. In this way, we study some particular objects as simple algebras, directly irreducible algebras, injectives, etc. A completeness theorem with respect to the standard MV-algebra, considered as σ-complete MV-algebra, is obtained. Finally, we apply this result to the study of σ-complete Boolean algebras and σ-complete product MV-algebras.