Lattice-ordered Groups Research Articles

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.

Generalized commutativity of lattice-ordered groups II

Commutative \({\ell}\)-groups G (in which for all \({x, y \in G, xy = yx}\)) were studied long ago. This was then generalized to the study of \({\ell}\)-groups G in which for a given integer n and for all \({x, y \in G, x^{n}y^{n} = y^{n}x^{n}}\). It was then discovered that if for all \({x, y \in G}\), both \({x^{n}y^{n} = y^{n}x^{n}}\) and \({x^{m}y^{m} = y^{m}x^{m}}\) for two different integers m, n, then also \({x^{d}y^{d} = y^{d}x^{d}}\), where d is the greatest common divisor of m, n.

The Conrad Program: From ℓ-groups to algebras of logic

A number of research articles have established the significant role of lattice-ordered groups (ℓ-groups) in logic. The fact that underpins these studies is the realization that important algebras of logic may be viewed as ℓ-groups with a modal operator. These connections are just the tip of the iceberg. The purpose of the present article is to lay the groundwork for, and provide significant initial contributions to, the development of a Conrad type approach to the study of algebras of logic. The term Conrad Program refers to Paul Conrad's approach to the study of ℓ-groups, which analyzes the structure of individual or classes of ℓ-groups by primarily using strictly lattice theoretic properties of their lattices of convex ℓ-subgroups. The present article demonstrates that large parts of the Conrad Program can be profitably extended in the setting of e-cyclic residuated lattices – that is residuated lattices that satisfy the identity x\\e≈e/x. An indirect benefit of this work is the introduction of new tools and techniques in the study of algebras of logic, and the enhanced role of the lattice of convex subalgebras of a residuated lattice.

The left adjoint of Spec from a category of lattice-ordered groups

Let us write ℓGuf for the category whose objects are lattice-ordered abelian groups (l-groups for short) with a strong unit and finite prime spectrum endowed with a collection of Archimedean elements, one for each prime l-ideal, which satisfy certain properties, and whose arrows are l-homomorphisms with additional structure. In this paper we show that a functor which assigns to each object (A,uˆ)∈ℓGuf the prime spectrum of A, and to each arrow f:(A,uˆ)→(B,vˆ)∈ℓGuf the naturally induced p-morphism, has a left adjoint.

Scalar extensions for algebraic structures of Łukasiewicz logic

In this paper we study the tensor product for MV-algebras, the algebraic structures of Łukasiewicz ∞-valued logic. Our main results are: the proof that the tensor product is preserved by the categorical equivalence between the MV-algebras and abelian lattice-order groups with strong unit and the proof of the scalar extension property for semisimple MV-algebras. We explore consequences of these results for various classes of MV-algebras and lattice-ordered groups enriched with a product operation.

Structure Theorems of Specker Groups (I)

Specker groups are subgroups of group of all finite values sequences of integers. Fuchs ([5]) developed many theorems about Specker groups. Here I want to use lattice-ordered group approach to develop the lateral completion of Specker groups and Specker spaces. The main goals of this research paper are to prove theorem 2.9 and corollary 2.10.

The Riesz Hull of a Semisimple MV-Algebra

Abstract MV-algebras and Riesz MV-algebras are categorically equivalent to abelian lattice-ordered groups with strong unit and, respectively, with Riesz spaces (vector-lattices) with strong unit. A standard construction in the literature of lattice-ordered groups is the vector-lattice hull of an archimedean latticeordered group. Following a similar approach, in this paper we define the Riesz hull of a semisimple MV-algebra.

The Chinese Remainder Theorem for Strongly Semisimple MV-Algebras and Lattice-Groups

Abstract An MV-algebra (equivalently, a lattice-ordered Abelian group with a distinguished order unit) is strongly semisimple if all of its quotients modulo finitely generated congruences are semisimple. All MV-algebras satisfy a Chinese Remainder Theorem, as was first shown by Keimel four decades ago in the context of lattice-groups. In this note we prove that the Chinese Remainder Theorem admits a considerable strengthening for strongly semisimple structures.

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.

Pointfree pointwise suprema in unital archimedean ℓ-groups

We generalize the concept of the pointwise supremum of real-valued functions to the pointfree setting. The concept itself admits a direct and intuitive formulation which makes no mention of points. But our aim here is to investigate pointwise suprema of subsets of RL, the family of continuous real valued functions on a locale, or pointfree space.Thus our setting is the category W of archimedean lattice-ordered groups (ℓ-groups) with designated weak order unit, with morphisms which preserve the group and lattice operations and take units to units. This is an appropriate context for this investigation because every W-object can be canonically represented as a subobject of some RL.We show that the suprema which are pointwise in the Madden representation can be characterized purely algebraically. They are precisely the suprema which are context-free, in the sense of being preserved by every W homomorphism out of G. We show that closure under such suprema characterizes the W-kernels among the convex ℓ-subgroups. Finally, we prove that all existing joins in a W-object G are pointwise iff its Madden frame L is boolean, and that all existing countable joins in G are pointwise if L is a P-frame, but not conversely.This leads up to the appropriate analog of the Nakano–Stone Theorem: a (completely regular) locale L has the feature that RL is conditionally pointwise complete (σ-complete), i.e., every bounded (countable) family from RL has a pointwise supremum in RL, iff L is boolean (a P-locale).We adopt a maximally broad definition of unconditional pointwise completeness (σ-completeness): a divisible W-object G is pointwise complete (σ-complete) if it contains a pointwise supremum for every subset which has a supremum in any extension. We show that the pointwise complete (σ-complete) W-objects are those of the form RL for L a boolean locale (P-locale). Finally, we show that a W-object G is pointwise σ-complete iff it is epicomplete.

Generalized Commutativity of Lattice-Ordered Groups

Abstract In this paper we explore generalizations of Neumann’s theorem proving that weak commutativity in ordered groups actually implies the group is abelian. We show that a natural generalization of Neumann’s weak commutativity holds for certain Scrimger ℓ-groups.

Bounded Equivalence of Hull Classes in Archimedean Lattice-Ordered Groups with Unit

A hull class in a category is an object class H for which each object has a unique minimal essential extension in H. This paper addresses the enormity of the collection of hull classes in the category W of Archimedean l-groups with distinguished weak order unit through consideration of the action on the hull classes of the bounded coreflection \(\textbf {W} \overset {B}\rightarrow \textbf {W}^{\ast }\) onto the subcategory where the units are strong. It is shown that hull classes go forth under B and back under B−1, that the B-equivalence class of a hull class in W always has a top, and that these B-equivalence classes are frequently not sets. The property “top” is related to various other properties that hull classes might have. This paper is the third by us on the complex taxonomy of hull classes in W, and more are planned.

Some unusual epicomplete Archimedean lattice-ordered groups

An Archimedean l l -group is epicomplete if it is divisible and σ \sigma -complete, both laterally and conditionally. Under various circumstances it has been shown that epicompleteness implies the existence of a compatible reduced f f -ring multiplication; the question has arisen whether or not this is always true. We show that a set-theoretic condition weaker than the continuum hypothesis implies “not”, and conjecture the converse. The examples also fail decent representation and existence of some other compatible operations.

Locally solid topological lattice-ordered groups

Locally solid Riesz spaces have been widely investigated in the past several decades; but locally solid topological lattice-ordered groups seem to be largely unexplored. The paper is an attempt to initiate a relatively systematic study of locally solid topological lattice-ordered groups. We give both Roberts-Namioka-type characterization and Fremlin-type characterization of locally solid topological lattice-ordered groups. In particular, we show that a group topology on a lattice-ordered group is locally solid if and only if it is generated by a family of translation-invariant lattice pseudometrics. We also investigate (1) the basic properties of lattice group homomorphism on locally solid topological lattice-ordered groups; (2) the relationship between order-bounded subsets and topologically bounded subsets in locally solid topological lattice-ordered groups; (3) the Hausdorff completion of locally solid topological lattice-ordered groups.

A note on a result of I. Gusić on two inequalities in lattice-ordered groups

A note on a result of I. Gusić on two inequalities in lattice-ordered groups

Truncated abelian lattice-ordered groups II: the pointfree (Madden) representation

This is the second of three articles on the topic of truncation as an operation on divisible abelian lattice-ordered groups, or simply ℓ-groups. This article uses the notation and terminology of the first article and assumes its results. In particular, we refer to an ℓ-group with truncation as a truncated ℓ-group, or simply a trunc, and denote the category of truncs with truncation homomorphisms by AT.Here we develop the analog for AT of Madden's pointfree representation for W, the category of archimedean ℓ-groups with designated order unit. More explicitly, for every archimedean trunc A there is a regular Lindelöf frame L equipped with a designated point ⁎:L→2, a subtrunc Aˆ of R0L, the trunc of pointed frame maps O0R→L, and a trunc isomorphism A→Aˆ. A pointed frame map is just a frame map between frames which commutes with their designated points, and O0R stands for the pointed frame which is the topology OR of the real numbers equipped with the frame map of the insertion 0→R. (L,⁎) is unique up to pointed frame isomorphism with respect to its properties. Finally, we reprove an important result from the first article, namely that W is a non-full monoreflective subcategory of AT.

Amalgamation bases for the class of lattice-ordered groups

Abstract We prove TheoremA. The cardinal product of two copies of the integers is an amalgamation base for the class of all lattice-ordered groups but their lexicographic product is not. This answers Problem 27 of [Black Swamp Problem Book (W. Charles Holland, ed.), Bowling Green State University, 1982]. We also prove TheoremB. he cardinal product of n copies of the integers is not an amalgamation base for the class of all lattice-ordered groups if n ≥ 3.

Hölder categories

Abstract Hölder categories are invented to provide an axiomatic foundation for the study of categories of archimedean lattice-ordered algebraic structures. The basis of such a study is Hölder’s Theorem (1908), stating that the archimedean totally ordered groups are precisely the subgroups of the additive real numbers ℝ with the usual addition and ordering, which remains the single most consequential result in the studies of lattice-ordered algebraic systems since Birkhoff and Fuchs to the present. This study originated with interest in W*, the category of all archimedean lattice-ordered groups with a designated strong order unit, and the ℓ-homomorphisms which preserve those units, and, more precisely, with interest in the epireflections on W*. In the course of this study, certain abstract notions jumped to the forefront. Two of these, in particular, seem to have been mostly overlooked; some notion of simplicity appears to be essential to any kind of categorical study of W*, as are the quasi-initial objects in a category. Once these two notions have been brought into the conversation, a Hölder category may then be defined as one which is complete, well powered, and in which (a) the initial object I is simple, and (b) there is a simple quasi-initial coseparator R. In this framework it is shown that the epireflective hull of R is the least monoreflective class. And, when I = R — that is, the initial element is simple and a coseparator — a theorem of Bezhanishvili, Morandi, and Olberding, for bounded archimedean f-algebras with identity, can be be generalized, as follows: for any Hölder category subject to the stipulation that the initial object is a simple coseparator, every uniformly nontrivial reflection — meaning that the reflection of each non-terminal object is non-terminal — is a monoreflection. Also shown here is the fact that the atoms in the class of epireflective classes are the epireflective hulls of the simple quasi-initial objects. From this observation one easily deduces a converse to the result of Bezhanishvili, Morandi, and Olberding: if in a Hölder category every epireflection is a monoreflection, then the initial object is a coseparator.

Lattice-theoretic properties of algebras of logic

In the theory of lattice-ordered groups, there are interesting examples of properties — such as projectability — that are defined in terms of the overall structure of the lattice-ordered group, but are entirely determined by the underlying lattice structure. In this paper, we explore the extent to which projectability is a lattice-theoretic property for more general classes of algebras of logic. For a class of integral residuated lattices that includes Heyting algebras and semi-linear residuated lattices, we prove that a member of such is projectable iff the order dual of each subinterval [a,1] is a Stone lattice. We also show that an integral GMV algebra is projectable iff it can be endowed with a positive Gödel implication. In particular, a ΨMV or an MV algebra is projectable iff it can be endowed with a Gödel implication. Moreover, those projectable involutive residuated lattices that admit a Gödel implication are investigated as a variety in the expanded signature. We establish that this variety is generated by its totally ordered members and is a discriminator variety.