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.     

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.     

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.     

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.     

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.     

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.     

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.     

Implication in MV-algebras

We set up an axiomatic system for the logical connective implication within the framework of MV-algebras which generalizes implication in classical logic described similarly by J. C. Abbott. The induced structure (weak implication algebra) turns to be a join-semilattice whose principal filters are MV-algebras.Received October 2, 2003; accepted in final form March 17, 2004.     

On some classes of state-morphism MV-algebras

Flaminio and Montagna recently introduced state MV-algebras as MV-algebras with an internal notion of a state. The present authors gave a stronger version of state MV-algebras, called state-morphism MV-algebras. We present some classes of state-morphism MV-algebras like local, simple, semisimple state-morphism MV-algebras, and state-morphism MV-algebras with retractive ideals. Finally, we describe state-morphism operators on m-free generated MV-algebras, m < ∞.     

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.     

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.     

Lattice generation of small equivalences of a countable set

Given a countable set A, let Equ(A) denote the lattice of equivalences of A. We prove the existence of a four-generated sublattice Q of Equ(A) such that Q contains all atoms of Equ(A). Moreover, Q can be generated by four equivalences such that two of them are comparable. Our result is a reasonable generalization of Strietz [5, 6] from the finite case to the countable one; and in spite of its essentially simpler proof it asserts more for the countable case than [2, 3].Dedicated to George Grätzer on his 60th birthdayThis research was supported by the NFSR of Hungary (OTKA), grant no. T7442.     

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.     

A note on essentially convex orders

In this note, we determine precisely which partially ordered sets (posets) have the property that, whenever they occur as subposets of a larger poset, they occur there convexly, i.e., as convex subposets. As a corollary, we also determine which lattices have the property that, if they occur as sublattices of a finite distributive lattice L, then they also occur as closed intervals in L. Throughout, all sets will be finite.Dedicated to the memory of Ivan RivalReceived May 5, 2003; accepted in final form October 3, 2004.This revised version was published online in August 2005 with a corrected cover date.     

Cores and retracts

Theretracts (idempotent, isotone self-maps) of an ordered set are naturally ordered as functions. In this note we characterize the possible ways that one retract can cover another one. This gives some insight into the structure of the ordered set of retracts and leads to a natural generalization of the core of an ordered set.Supported by NSERC Operating Grant 41702.     

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     

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.     

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.     

A chain complete poset with no infinite antichain has a finite core

It is well known that dismantling a finite posetP leads to a retract, called the core ofP, which has the fixed-point property if and only ifP itself has this property. The PT-order, or passing through order, of a posetP is the quasi order defined onP so thatab holds if and only if every maximal chain ofP which passes througha also passes throughb. This leads to a generalization of the dismantling procedure which works for arbitrary chain complete posets which have no infinite antichain. We prove that such a poset also has a finite core, i.e. a finite retract which reflects the fixed-point property forP.This research was written while the first author was visiting the University of Calgary.Research supported by NSERC grant #69-0982.     

Many varieties of lattices with nonsurjective epimorphisms

We use dominions to show that many varieties of lattices have nonsurjective epimorphisms. The variety D of distributive lattices is treated in detail. We show that the dominion in D of a sublattice is the closure of M under relative complementation in L. This dominion is also the largest sublattice of L in which M is epimorphically embedded. In any variety of lattices larger than D, the dominion of M in L is just M. Received May 1, 2001; accepted in final form October 4, 2005.