Abelian L-groups Research Articles

A solution to the MV-spectrum problem in size aleph one

Denote by IdcG the lattice of all principal ℓ-ideals of an Abelian ℓ-group G. Our main result is the following. TheoremFor every countable Abelian ℓ-group G, every countable completely normal distributive 0-lattice L, and every closed 0-lattice homomorphismφ:IdcG→L, there are a countable Abelian ℓ-group H, an ℓ-homomorphismf:G→H, and a lattice isomorphismι:IdcH→Lsuch thatφ=ι∘Idcf. We record the following consequences of that result:(1)A 0-lattice homomorphism φ:K→L, between countable completely normal distributive 0-lattices, can be represented, with respect to the functor Idc, by an ℓ-homomorphism of Abelian ℓ-groups iff it is closed.(2)A distributive 0-lattice D of cardinality at most ℵ1 is isomorphic to some IdcG iff D is completely normal and for all a,b∈D the set {x∈D|a≤b∨x} has a countable coinitial subset. This solves Mundici's MV-spectrum Problem for cardinalities up to ℵ1. The bound ℵ1 is sharp.Item (1) is extended to commutative diagrams indexed by forests in which every node has countable height. All our results are stated in terms of vector lattices over any countable totally ordered division ring.

Complemented MacNeille completions and algebras of fractions

We introduce (ℓ-)bimonoids as ordered algebras consisting of two compatible monoidal structures on a partially ordered (lattice-ordered) set. Bimonoids form an appropriate framework for the study of a general notion of complementation, which subsumes both Boolean complements in bounded distributive lattices and multiplicative inverses in monoids. The central question of the paper is whether and how bimonoids can be embedded into complemented bimonoids, generalizing the embedding of cancellative commutative monoids into their groups of fractions and of bounded distributive lattices into their free Boolean extensions. We prove that each commutative (ℓ-)bimonoid embeds into a complete complemented commutative ℓ-bimonoid in a doubly dense way reminiscent of the Dedekind–MacNeille completion. Moreover, this complemented completion, which is term equivalent to a commutative involutive residuated lattice, sometimes contains a tighter complemented envelope analogous to the group of fractions. In the case of cancellative commutative monoids this algebra of fractions is precisely the familiar group of fractions, while in the case of Brouwerian (Heyting) algebras it is a (bounded) idempotent involutive commutative residuated lattice. This construction of the algebra of fractions in fact yields a categorical equivalence between varieties of integral and of involutive residuated structures which subsumes as special cases the known equivalences between Abelian ℓ-groups and their negative cones, and between Sugihara monoids and their negative cones.

From distributive ℓ-monoids to ℓ-groups, and back again

We prove that an inverse-free equation is valid in the variety LG of lattice-ordered groups (ℓ-groups) if and only if it is valid in the variety DLM of distributive lattice-ordered monoids (distributive ℓ-monoids). This contrasts with the fact that, as proved by Repnitskiĭ, there exist inverse-free equations that are valid in all Abelian ℓ-groups but not in all commutative distributive ℓ-monoids, and, as we prove here, there exist inverse-free equations that are valid in all totally ordered groups but not in all totally ordered monoids. We also prove that DLM has the finite model property and a decidable equational theory, establish a correspondence between the validity of equations in DLM and the existence of certain right orders on free monoids, and provide an effective method for reducing the validity of equations in LG to the validity of equations in DLM.

Cevian properties in ideal lattices of Abelian ℓ-groups

Abstract We consider the problem of describing the lattices of compact ℓ {\ell} -ideals of Abelian lattice-ordered groups. (Equivalently, describing the spectral spaces of Abelian lattice-ordered groups.) It is known that these lattices have countably based differences and admit a Cevian operation. Our first result says that these two properties are not sufficient: there are lattices having both countably based differences and Cevian operations, which are not representable by compact ℓ {\ell} -ideals of Abelian lattice-ordered groups. As our second result, we prove that every completely normal distributive lattice of cardinality at most ℵ 1 {\aleph_{1}} admits a Cevian operation. This complements the recent result of F. Wehrung, who constructed a completely normal distributive lattice having countably based differences, of cardinality ℵ 2 {\aleph_{2}} , without a Cevian operation.

Boolean lifting property in quantales

In ring theory, the lifting idempotent property (LIP) is related to some important classes of rings: clean rings, exchange rings, local and semilocal rings, Gelfand rings, maximal rings, etc. Inspired by LIP, lifting properties were also defined for other algebraic structures: MV-algebras, BL-algebras, residuated lattices, abelian l-groups, congruence distributive universal algebras, etc. In this paper, we define a lifting property (LP) in commutative coherent integral quantales, structures that are a good abstraction for lattices of ideals, filters and congruences. LP generalizes all the lifting properties existing in the literature. The main tool in the study of LP in a quantale A is the reticulation of A, a bounded distributive lattice whose prime spectrum is homeomorphic to the prime spectrum of A. The principal results of the paper include a characterization theorem for quantales with LP and a characterization theorem for hyperarchimedean quantales.

Cevian operations on distributive lattices

We construct a completely normal bounded distributive lattice D in which for every pair (a,b) of elements, the set {x∈D|a≤b∨x} has a countable coinitial subset, such that D does not carry any binary operation ∖ satisfying the identities x≤y∨(x∖y), (x∖y)∧(y∖x)=0, and x∖z≤(x∖y)∨(y∖z). In particular, D is not a homomorphic image of the lattice of all finitely generated convex ℓ-subgroups of any (not necessarily Abelian) ℓ-group. It has ℵ2 elements. This solves negatively a few problems stated by Iberkleid, Martínez, and McGovern in 2011 and recently by the author. This work also serves as preparation for a forthcoming paper in which we prove that for any infinite cardinal λ, the class of Stone duals of spectra of all Abelian ℓ-groups with order-unit is not closed underL∞λ-elementary equivalence.

A weighted Möbius function

Fix an odd prime ℓ and let G be the poset of isomorphism classes of finite abelian ℓ-groups, ordered by inclusion. If ξ:G→R≥0 is a discrete probability distribution on G and A∈G, define the Ath moment of ξ to be ∑B∈G|Surj(B,A)|ξ(B). The question of determining conditions that ensure ξ is completely determined by its moments has been of recent interest in many problems of Cohen–Lenstra type. Furthermore, recovering ξ from its moments requires a new Möbius-type inversion formula on G. In this paper, we define this function, relate it to the classical Möbius function on the poset of subgroups of a fixed group, and prove two theorems about when it vanishes. As one corollary of these theorems, we obtain an analog of Hall’s theorem on the vanishing of the classical Möbius function. As another, we obtain an infinite family of pairs of groups on which the classical Möbius function vanishes; obtaining such pairs is a group-theoretic topic of recent interest.

Idempotent endomorphisms of free MV-algebras and unital ℓ-groups

Several papers deal with the problem of counting the number of idempotent endomorphisms of a structure S onto a substructure T. In this paper we consider the case when T is a projective lattice-ordered abelian group with a distinguished strong order unit, or equivalently, a projective MV-algebra. Suppose A is the image (=range) of an idempotent endomorphism of the free n-generator MV-algebra M([0,1]n) of McNaughton functions on [0,1]n. We prove that the number r(A) of idempotent endomorphisms of M([0,1]n) onto A is finite if, and only if, the maximal spectral space μA is homeomorphic to a (Kuratowski) closed domain M of [0,1]n, in the sense that M=cl(int(M)). Further, the closed domain condition is decidable and r(A) is computable, once an idempotent endomorphism of M([0,1]n) onto A is explicitly given. Thus every finitely generated projective MV-algebra B comes equipped with a new invariant ι(B)=sup⁡{r(A)|A≅B, for A the image of an idempotent endomorphism of M([0,1]k)}, and k the smallest number of generators of B. We compute ι(B) for many projective MV-algebras B existing in the literature. Various problems concerning idempotent endomorphisms of free MV-algebras are shown to be decidable. Via the Γ functor, our results and computations automatically transfer to finitely generated projective abelian ℓ-groups with a distinguished strong unit.

Algebraically closed and existentially closed Abelian lattice-ordered groups

If \({\mathcal{G}}\) is an Abelian lattice-ordered (l-) group, then \({\mathcal{G}}\) is algebraically (existentially) closed just in case every finite system of l-group equations (equations and inequations), involving elements of \({\mathcal{G}}\), that is solvable in some Abelian l-group extending \({\mathcal{G}}\) is solvable already in \({\mathcal{G}}\). This paper establishes two systems of axioms for algebraically (existentially) closed Abelian l-groups, one more convenient for modeltheoretic applications and the other, discovered by Weispfenning, more convenient for algebraic applications. Among the model-theoretic applications are quantifierelimination results for various kinds of existential formulas, a new proof of the amalgamation property for Abelian l-groups, Nullstellensatze in Abelian l-groups, and the display of continuum-many elementary-equivalence classes of existentially closed Archimedean l-groups. The algebraic applications include demonstrations that the class of algebraically closed Abelian l-groups is a torsion class closed under arbitrary products, that the class of l-ideals of existentially closed Abelian l-groups is a radical class closed under binary products, and that various classes of existentially closed Abelian l-groups are closed under bounded Boolean products.

Algebraically Closed Abelian l-Groups

Abstract Every semifield of non-zero characteristic is either a field of prime characteristic or a semifield of characteristic 1. Semifields of characteristic 1 are equivalent to abelian lattice-ordered groups. It is proved that such a semifield A is algebraically closed if and only if the pure equations xn = a and certain quadratic equations are solvable in A. Using a sheaf representation for z-projectable abelian l-groups on the co-Zariski space of minimal primes, a sheaf-theoretic characterization of algebraic closedness in characteristic 1 is obtained. Concerning the solvability of quadratic equations, the criterion consists in a topological condition for the base space. The results are built upon an analysis of rational functions in characteristic 1. While polynomials satisfy the “fundamental theorem of algebra”, the multiplicative structure of rational functions is determined by means of “divisors”, extracted from the additive structure of A modulo parallelogram identities.

Right l-groups, geometric Garside groups, and solutions of the quantum Yang–Baxter equation

Right lattice-ordered groups are introduced and studied as a general framework for Garside groups and related groups without a Garside element. Every right l-group G has a (two-sided) partially ordered subgroup N(G) which generalizes the quasi-centre of an Artin–Tits group. The group N(G) splits into copies of Z if G is noetherian. The positive cone of a right l-group is described as a structure that is known from algebraic logic: a pair of left and right self-similar hoops. For noetherian right l-groups G, modularity of the lattice structure is characterized in terms of an operation on the set X(G−) of atoms. It is proved that modular Garside groups are equivalent to finite projective spaces with a non-degenerate labelling. A concept of duality for X(G−) is introduced and applied in the distributive case. This gives a one-to-one correspondence between noetherian right l-groups with duality and non-degenerate unitary set-theoretic solutions of the quantum Yang–Baxter equation. The description of Garside groups via Garside germs is extended to right l-groups, which yields a one-sided enhancement and a new proof of Dvurečenskij's non-commutative extension of Mundici's correspondence between abelian l-groups and MV-algebras.

The Morita-equivalence between MV-algebras and lattice-ordered abelian groups with strong unit

We show that the theory of MV-algebras is Morita-equivalent to that of abelian ℓ-groups with strong unit. This generalizes the well-known equivalence between the categories of set-based models of the two theories established by D. Mundici in 1986, and allows to transfer properties and results across them by using the methods of topos theory. We discuss several applications, including a sheaf-theoretic version of Mundici's equivalence and a bijective correspondence between the geometric theory extensions of the two theories.

From Interior Algebras to Unital ℓ-Groups: A Unifying Treatment of Modal Residuated Lattices

Much work has been done on specific instances of residuated lattices with modal operators (either nuclei or conuclei). In this paper, we develop a general framework that subsumes three important classes of modal residuated lattices: interior algebras, Abelian l-groups with conuclei, and negative cones of l-groups with nuclei. We then use this framework to obtain results about these three cases simultaneously. In particular, we show that a categorical equivalence exists in each of these cases. The approach used here emphasizes the role played by reducts in the proofs of these categorical equivalences. Lastly, we develop a connection between translations of logics and images of modal operators.

Varieties generated by unital Abelian ℓ-groups

Much work has been done on the connections between Abelian ℓ-groups and MV-algebras. For example, it is well known that there is a bijective correspondence between the so-called equational classes of unital Abelian ℓ-groups and the subvarieties of MV, the variety of MV-algebras. The drawback of this correspondence is that these former classes are not varieties. The main result of this paper remedies this situation by establishing that the subvariety lattice of the variety pA of positively-pointed Abelian ℓ-groups (excluding the trivial variety) is isomorphic to the subvariety lattice of MV. In particular, with the exception of the unique atom, every subvariety of pA is generated by its unital members.

Abelian lattice-ordered groups and a characterization of the maximal spectrum of a Prüfer domain

The concept of z-projectable abelian lattice-ordered group is introduced, and it is shown that every such group G can be identified with the group of global sections of a sheaf G with totally ordered stalks on the co-Zariski space MinG of minimal prime ideals. Semi-projectable abelian l-groups are z-projectable, but not vice versa. The sheaves G as well as the spaces MinG arising from abelian l-groups G are characterized completely. Using Hochster duality and the Jaffard–Ohm correspondence, the results are applied to determine the maximal spectrum of a Prüfer domain and of a Bézout domain.

Projective objects in the categories of abelian ℓ-groups and MV-algebras

We provide a description of free MV-algebras as subalgebras of intervals of free Abelian l-groups. This description shows that the unital Abelian l-group corresponding to a free MV-algebra is projective as an Abelian l-group. More generally, we prove that this is still the case for projective MV-algebras. Using a construction similar to the one for MV-algebras, we show that the free algebras in the variety of negative cones of Abelian l-groups are subalgebras of negative cones of free Abelian l-groups. This allows us to prove a Baker–Beynon-type theorem for finitely generated free algebras in the variety of negative cones of Abelian l-groups. These results are specific cases of a more general situation, which is the subject of the last section of this paper.

Radical Classes Closed Under Products

Abstract This is a survey of what is known about Kurosh-Amitsur radical classes which are closed under direct products. Associative rings, groups, abelian groups, abelian ℓ-groups and modules are treated. We have attempted to account for all published results relevant to this topic. Many or most of these were not, as published, expressed in radical theoretic terms, but have consequences for radical theory which we point out. A fruitful source of results and examples is the notion of slenderness for abelian groups together with its several variants for other structures. We also present a few new results, including examples and a demonstration that e-varieties of regular rings are product closed radical classes of associative rings.

A Note on Bosbach’s Cone Algebras

In 2002, Dvure?enskij extended Mundici's equivalence between unital abelian l-groups and MV-algebras to the non-commutative case. We analyse the relationship to Bosbach's cone algebras and clarify the role of the corresponding pair of L-algebras. As a consequence, it follows that one of the two L-algebra axioms can be dropped.

Topologies arising from metrics valued in abelian ℓ-groups

This paper considers metrics valued in abelian l-groups and their induced topologies. In addition to a metric into an l-group, one needs a filter in the positive cone to determine which balls are neighborhoods of their center. As a key special case, we discuss a topology on a lattice ordered abelian group from the metric dG and the positive filter consisting of the weak units of G; in the case of \({\mathbb R^{n}}\) , this is the Euclidean topology. We also show that there are many Nachbin convex topologies on an l-group which are not induced by any positive filter of the l-group.

From free abelian groups to free abelian ℓ-groups

Abstract For each n, the free abelian group F on countably many free generators will be equipped with a monoid P n making (F,P n) into the free n-generator abelian ℓ-group. This is a generalization of the main result of the second author, published in the J. Pure Appl. Algebra 208 (2007), 549–554, only dealing with the case n = 2.