State-morphism MV-algebras

We present a stronger variation of state MV-algebras, recently presented by T. Flaminio and F. Montagna, which we call state-morphism MV-algebras. Such structures are MV-algebras with an internal notion, a state-morphism operator. We describe the categorical equivalences of such (state-morphism) state MV-algebras with the category of unital Abelian ?-groups with a fixed state operator and present their basic properties. In addition, in contrast to state MV-algebras, we are able to describe all subdirectly irreducible state-morphism MV-algebras.     

Bounded pseudo-hoops with internal states

State MV-algebras were introduced by Flaminio and Montagna as MV-algebras with internal states. Di Nola and Dvure?enskij presented the notion of state-morphism MV-algebra which is a stronger variation of a state MV-algebra. Rach?nek and ?alounová introduced state GMV-algebras (pseudo-MV algebras) and state-morphism GMV-algebras, while the state BL-algebras and state-morphism BL-algebras were defined by Ciungu, Dvure?enskij and Hy?ko. Recently, Dvure?enskij, Rach?nek and ?alounová presented state R?-monoids and state-morphism R?-monoids. In this paper we study these concepts for more general fuzzy structures, namely pseudo-hoops and we present state pseudo-hoops and state-morphism pseudo-hoops.     

Subdirectly irreducible state-morphism BL-algebras

Recently Flaminio and Montagna (Proceedings of the 5th EUSFLAT Conference, II: 201?C206. Ostrava, 2007), (Inter. J. Approx. Reason. 50:138?C152, 2009) introduced the notion of a state MV-algebra as an MV-algebra with internal state. We have two kinds: state MV-algebras and state-morphism MV-algebras. These notions were also extended for state BL-algebras in (Soft Comput. doi:10.1007/s00500-010-0571-5). In this paper, we completely describe subdirectly irreducible state-morphism BL-algebras and this generalizes an analogous result for state-morphism MV-algebras presented in (Ann. Pure Appl. Logic 161:161?C173, 2009).     

On varieties of MV-algebras with internal states

In [4] and [5] the authors introduced the variety SMV of MV-algebras with an internal operator, state MV-algebras. In [2] and [3] the authors gave a stronger version of state MV-algebras, called state-morphism MV-algebras. In this paper we continue the studies presented in [2] and [3] just looking at several proper subvarieties of SMV, obtained by imposing suitable conditions on the behavior of the internal operator.     

Erratum to “State-morphism MV-algebras” [Ann. Pure Appl. Logic 161 (2009) 161-173]

Recently, the first two authors characterized in Di Nola and Dvure?enskij (2009) [1] subdirectly irreducible state-morphism MV-algebras. Unfortunately, the main theorem (Theorem 5.4(ii)) has a gap in the proof of Claim 10, as the example below shows. We now present a correct characterization and its correct proof.     

Weak MV-algebras

In a recent paper [CHAJDA, I.—KüHR, J.: A non-associative generalization of MV-algebras, Math. Slovaca 57, (2007), 301–312], authors introduced and studied a non-associative generalization of MV-algebras called NMV-algebras. In contrast to MV-algebras, sections (i.e. principal filters) in NMV-algebras which are proper (i.e. are not MV-algebras), do not admit a structure of an NMV-algebra with respect to the operations defined in a natural way. The aim of the paper is to present a new class of algebras generalizing MV-algebras but sharing the above property. The financial support by the grant of Czech Government MSM 6198959214 is gratefully acknowledged.     

On the structure of rotation-invariant semigroups

 We generalize the notions of Girard algebras and MV-algebras by introducing rotation-invariant semigroups. Based on a geometrical characterization, we present five construction methods which result in rotation-invariant semigroups and in particular, Girard algebras and MV-algebras. We characterize divisibility of MV-algebras, and point out that integrality of Girard algebras follows from their other axioms. Received: 7 January 2002 / Revised version: 4 April 2002 / Published online: 19 December 2002 RID="*" ID="*" Supported by the National Scientific Research Fund Hungary (OTKA F/032782). Mathematics Subject Classification (2000): 20M14, 06F05 Key words or phrases: Residuated lattice – Conjunction for non-classical logics     

Representation of MV-algebras by regular ultrapowers of [0, 1]

We present a uniform version of Di Nola Theorem, this enables to embed all MV-algebras of a bounded cardinality in an algebra of functions with values in a single non-standard ultrapower of the real interval [0,1]. This result also implies the existence, for any cardinal α, of a single MV-algebra in which all infinite MV-algebras of cardinality at most α embed. Recasting the above construction with iterated ultrapowers, we show how to construct such an algebra of values in a definable way, thus providing a sort of “canonical” set of values for the functional representation.     

A common generalization for MV-algebras and Łukasiewicz–Moisil algebras

We introduce the notion of n-nuanced MV-algebra by performing a Łukasiewicz–Moisil nuancing construction on top of MV-algebras. These structures extend both MV-algebras and Łukasiewicz–Moisil algebras, thus unifying two important types of structures in the algebra of logic. On a logical level, n-nuanced MV-algebras amalgamate two distinct approaches to many valuedness: that of the infinitely valued Łukasiewicz logic, more related in spirit to the fuzzy approach, and that of Moisil n-nuanced logic, which is more concerned with nuances of truth rather than truth degree. We study n-nuanced MV-algebras mainly from the algebraic and categorical points of view, and also consider some basic model-theoretic aspects. The relationship with a suitable notion of n-nuanced ordered group via an extension of the Γ construction is also analyzed.     

Abelian <Emphasis Type="Italic">ℓ</Emphasis>-Groups with Strong Unit and Perfect MV-Algebras

We investigate the class of abelian ℓ-groups with strong unit corresponding to perfect MV-algebras via the Γ functor, showing that this is a universal subclass of the class of all abelian ℓ-groups with strong unit and describing the formulas that axiomatize it. We further describe results for classes of abelian ℓ-groups with strong unit corresponding to local MV-algebras with finite rank.     

On direct limits of MV-algebras

We prove that the correspondence between MV-algebras and abelian ℓ-groups with the strong unit is preserved in the direct limit construction. Further, several classes of MV-algebras which are closed under formation of direct limits will be distinguished.     

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.     

Covers of the Abelian Variety of Generalized MV-Algebras

Using the categorical equivalence of the class of generalized MV-algebras with the class of unital ?-groups, we describe all varieties of symmetric top abelian unital ?-groups that cover the variety  _u? of abelian unital ?-groups. Equivalently, we describe all cover varieties of the variety of MV-algebras, ?, within the variety of generalized MV-algebras admitting only one negation and each of whose maximal ideals is normal. In particular, there are continuum many representable varieties of generalized MV-algebras that cover ?.     

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.     

态R_0代数

本文研究了R_0代数上有关态算子的问题.利用MV-代数上内态的引入方法引入了态算子,定义了态R_0代数,它是R_0代数的一般化.给出了一些非平凡态R_0代数的例子并讨论了态R_0代数的一些基本性质.在此基础上给出了态滤子和态局部R_0代数的概念,并利用态滤子刻画了态局部R_0代数.推广了局部R_0代数的相关理论.     

Non-commutative Łukasiewicz propositional logic

The non-commutative counterpart of the well-known Łukasiewicz propositional logic is developed, in strong connection with the algebraic theory of psMV-algebras. An extension by a new unary logical connective is also considered and a stronger completeness result is proved for this system.     

State operators on generalizations of fuzzy structures

We enlarge the language of R?-monoids, which are a non-commutative generalizations of both MV algebras and BL algebras, by adding a unary operation that describes algebraic properties of a state (= an analog of probability measures). The resulting algebras are called stateR?-monoids and state-morphismR?-monoids. We present basic properties of such algebras. We describe subdirectly irreducible algebras, some generators of the varieties of state-morphism R?-monoids, and an interplay between states and state operators.     

MV-algebras with operators (the commutative and the non-commutative case)

In the present paper we define the (pseudo) MV-algebras with n-ary operators, generalizing MV-modules and product MV-algebras. Our main results assert that there are bijective correspondences between the operators defined on a pseudo MV-algebra and the operators defined on the corresponding ℓ-group. We also provide a categorical framework and we prove the analogue of Mundici's categorical equivalence between MV-algebras and abelian ℓ-groups with strong unit. Thus, the category of pseudo MV-algebras with operators is equivalent to some category of ℓ-groups with operators.     

Free products of unital ℓ-groups and free products of generalized MV-algebras

We generalize Komori’s characterization of the proper subvarieties of MV-algebras. Namely, within the variety of generalized MV-algebras (GMV-algebras) such that every maximal ideal is normal, we characterize the proper top varieties. In addition, we present equational bases for these top varieties. We show that there are only countably many different proper top varieties and each of them has uncountably many subvarieties. Finally, we study coproducts and we show that the amalgamation property fails for the class of n-perfect GMV-algebras, i.e., GMV-algebras that can be split into n + 1 comparable slices. This paper has been supported by the Center of Excellence SAS -Physics of Information-I/2/2005, the grant VEGA No. 2/6088/26 SAV, by Science and Technology Assistance Agency under the contracts No. APVT-51-032002, APVV-0071-06, Bratislava.     

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.