Lattice-ordered Abelian Groups Research Articles

Revisiting the Farey AF Algebra

In a recent paper, F. Boca investigates the AF algebra \({{\mathfrak{A}}}\) associated with the Farey-Stern-Brocot sequence. We show that \({{\mathfrak{A}}}\) coincides with the AF algebra \({{\mathfrak{M_{1}}}}\) introduced by the present author in 1988. As proved in that paper (Adv. Math., vol.68.1), the K0-group of \({\mathfrak{A}}\) is the lattice-ordered abelian group \({\mathcal{M}_{1}}\) of piecewise linear functions on the unit interval, each piece having integer coefficients, with the constant 1 as the distinguished order unit. Using the elementary properties of \({\mathcal{M}_{1}}\) we can give short proofs of several results in Boca’s paper. We also prove many new results: among others, \({{\mathfrak{A}}}\) is a *-subalgebra of Glimm universal algebra, tracial states of \({{\mathfrak{A}}}\) are in one-one correspondence with Borel probability measures on the unit real interval, all primitive ideals of \({{\mathfrak{A}}}\) are essential. We describe the automorphism group of \({{\mathfrak{A}}}\) . For every primitive ideal I of \({{{\mathfrak{A}}}}\) we compute K0(I) and \({{K_{0}(\mathfrak{A}/I)}}\).

Finitely presented lattice-ordered abelian groups with order-unit

Let G be an ℓ-group (which is short for “lattice-ordered abelian group”). Baker and Beynon proved that G is finitely presented iff it is finitely generated and projective. In the category U of unital ℓ-groups, those ℓ-groups having a distinguished order-unit u, only the ( ⇐ ) -direction holds in general. We show that a unital ℓ-group ( G , u ) is finitely presented iff it has a basis. A large class of projectives is constructed from bases having special properties.

Boolean Skeletons of MV-algebras and ℓ-groups

Let Γ be Mundici's functor from the category $${\mathcal{LG}}$$ whose objects are the lattice-ordered abelian groups (l-groups for short) with a distinguished strong order unit and the morphisms are the unital homomorphisms, onto the category $${\mathcal{MV}}$$ of MV-algebras and homomorphisms. It is shown that for each strong order unit u of an l-group G, the Boolean skeleton of the MV-algebra Γ(G, u) is isomorphic to the Boolean algebra of factor congruences of G.

New torsion theory in unital abelian ℓ-groups

Abstract This paper introduces the notion of a functorial torsion class (FTC): in a concrete category $\mathfrak{C}$ which has image factorization, one considers monocoreflective subcategories which are closed under formation of subobjects. Here the interest is in FTCs in the category of abelian lattice-ordered groups with designated strong order unit. The FTCs $\mathfrak{T}$ consisting of archimedean latticeordered groups are characterized: for each subgroup A of the rationals with the identity 1, either $\mathfrak{T} = \mathfrak{S}\left( A \right)$, the class of all lattice-ordered groups of functions on a set X which have finite range in A, or $$\mathfrak{T} = \mathbb{T}\left( A \right)$$, the class of all subgroups of A with 1. As for FTCs possessing non-archimedean groups, it is shown that if $\mathfrak{T}$ is an FTC containing a subgroup A of the reals with 1, of rank two or greater, then $\mathfrak{T}$ contains all ℓ-groups of the form $A\vec \times G$, for all abelian lattice-ordered groups G. Finally, the least FTC that contains a non-archimedean group is the class of all $\mathbb{Z}\vec \times G$, for all abelian lattice-ordered groups G.

Divisibility theory of semi-hereditary rings

The semigroup of finitely generated ideals partially ordered by inverse inclusion, i.e., the divisibility theory of semi-hereditary rings, is precisely described by semi-hereditary Bezout semigroups. A Bezout semigroup is a commutative monoid S S with 0 such that the divisibility relation a | b ⟺ b ∈ a S a\vert b \Longleftrightarrow b\in aS is a partial order inducing a distributive lattice on S S with multiplication distributive on both meets and joins, and for any a , b , d = a ∧ b ∈ S , a = d a 1 a,\, b,\, d=a\wedge b\in S,\, a=da_1 there is b 1 ∈ S b_1\in S with a 1 ∧ b 1 = 1 , b = d b 1 a_1\wedge b_1=1,\, b=db_1 . S S is semi-hereditary if for each a ∈ S a\in S there is e 2 = e ∈ S e^2=e\in S with e S = a ⊥ = { x ∈ S | a x = 0 } eS=a^{\perp }=\{x\in S\,\vert \, ax=0\} . The dictionary is therefore complete: abelian lattice-ordered groups and semi-hereditary Bezout semigroups describe divisibility of Prüfer (i.e., semi-hereditary) domains and semi-hereditary rings, respectively. The construction of a semi-hereditary Bezout ring with a pre-described semi-hereditary Bezout semigroup is inspired by Stone’s representation of Boolean algebras as rings of continuous functions and by Gelfand’s and Naimark’s analogous representation of commutative C ∗ C^* -algebras.

Semigroup crossed products and the induced algebras of lattice-ordered groups

Let ( G , G + ) be a quasi-lattice-ordered group with positive cone G + . Laca and Raeburn have shown that the universal C ∗ -algebra C ∗ ( G , G + ) introduced by Nica is a crossed product B G + × α G + by a semigroup of endomorphisms. The goal of this paper is to extend some results for totally ordered abelian groups to the case of discrete lattice-ordered abelian groups. In particular given a hereditary subsemigroup H + of G + we introduce a closed ideal I H + of the C ∗ -algebra B G + . We construct an approximate identity for this ideal and show that I H + is extendibly α-invariant. It follows that there is an isomorphism between C ∗ -crossed products ( B G + / I H + ) × α ˜ G + and B ( G / H ) + × β G + . This leads to our main result that B ( G / H ) + × β G + is realized as an induced C ∗ -algebra Ind H ⊥ G ˆ ( B ( G / H ) + × τ ( G / H ) + ) .

Ordered groups with a conucleus

Our work proposes a new paradigm for the study of various classes of cancellative residuated lattices by viewing these structures as lattice-ordered groups with a suitable operator (a conucleus). One consequence of our approach is the categorical equivalence between the variety of cancellative commutative residuated lattices and the category of abelian lattice-ordered groups endowed with a conucleus whose image generates the underlying group of the lattice-ordered group. In addition, we extend our methods to obtain a categorical equivalence between Π MTL -algebras and product algebras with a conucleus. Among the other results of the paper, we single out the introduction of a categorical framework for making precise the view that some of the most interesting algebras arising in algebraic logic are related to lattice-ordered groups. More specifically, we show that these algebras are subobjects and quotients of lattice-ordered groups in a “quantale like” category of algebras.

The Lebesgue state of a unital abelian lattice-ordered group, II

We characterize the Lebesgue state of a free finitely generated unital lattice-ordered abelian group G in terms of its value at each element of each basis of G. This significantly strengthens one of the main results of our previous paper (co-authored by D. Mundici) with the same title as the present one. As a consequence of independent interest, we obtain a state-theoretic characterization of free finitely generated objects in the category of unital lattice-ordered abelian groups and their unit-preserving lattice-group homomorphisms.

Square means and geometric means in lattice-ordered groups and vector lattices

The geometric mean and the square mean of two positive real numbers are determined by the order in the sense that they can be written as (respectively) an infinite meet and an infinite join. We use this observation to define the geometric mean and the square mean on the positive elements of an Abelian lattice-ordered group. For certain totally ordered fields and vector lattices, the square mean can be used to define a new compatible addition. The resulting structure lives inside the complexification of the original structure and is constructed by using a general method for extending commutative lattice-ordered monoids to Abelian lattice-ordered groups. We describe this method and use it as the starting point for similar constructions of lattice-ordered rings and vector lattices.

Invariant Measures in Free MV-Algebras

MV-algebras can be viewed either as the Lindenbaum algebras of Łukasiewicz infinite-valued logic, or as unit intervals of lattice-ordered abelian groups in which a strong order unit has been fixed. The free n-generated MV-algebra Free n is representable as an algebra of continuous piecewise-linear functions with integer coefficients over the unit cube [0, 1] n . The maximal spectrum of Free n is canonically homeomorphic to [0, 1] n , and the automorphisms of the algebra are in 1–1 correspondence with the pwl homeomorphisms with integer coefficients of the unit cube. In this article, we prove that the only probability measure on [0, 1] n which is null on underdimensioned 0-sets and is invariant under the group of all such homeomorphisms is the Lebesgue measure. From the viewpoint of lattice-ordered abelian groups, this fact means that, in relevant cases, fixing an automorphism-invariant strong unit implies fixing a distinguished probability measure on the maximal spectrum. From the viewpoint of algebraic logic, it means that the only automorphism-invariant truth averaging process that detects pseudotrue propositions is the integral with respect to Lebesgue measure.

Weinberg's theorem, Elliott's ultrasimplicial property, and a characterisation of free lattice-ordered Abelian groups

We investigate the structure of lattice-preserving homomorphisms of free lattice-ordered Abelian groups to the ordered group of integers. For any lattice-ordered group, a choice of generators induces on such homomorphisms a partial commutative monoid canonically embedded into a direct product of the group of integers. Free lattice-ordered Abelian groups can be characterised in terms of this dual object and its embedding. For finite sets of generators, we obtain the stronger result: a lattice-ordered Abelian group is free on a finite generating set if and only if the generators make ℤ-valued homomorphisms a free Abelian group of finite rank. One of the main points of the paper is that all results are proved in an entirely elementary and self-contained manner. To achieve this end, we give a short new proof of the standard result of Weinberg that free lattice-ordered Abelian groups have enough ℤ-valued homomorphisms. The argument uses the ultrasimplicial property of ordered Abelian groups, first established by Elliott in a different connection. The paper is made self-contained by a new proof of Elliott's result.

The Haar theorem for lattice-ordered abelian groups with order-unit

Given a finite union $P$ of rational simplexes, we assign to $P$ numerical invariants $\lambda_{0}, \lambda_{1},\ldots,\lambda_{\dim P};$ each $\lambda_{i}$ is the suitably normalized volume of the $i$-dimensional part of $P$. We then prove that every finitely generated projective lattice-ordered abelian group $G$ with order-unit $u$ has a faithful invariant positive linear functional $s: G\to \mathbb R$. For each $g\in G$, $s(g)$ is the integral of $g$ over the maximal spectrum of $G$, the latter being canonically identified with a rational polyhedron $P$. Volume elements are measured by the $\lambda_{i}$'s. The proof uses the polyhedral versions of the Wlodarczyk-Morelli theorem on decompositions of birational toric maps in blow-ups and blow-downs, and of the De Concini-Procesi theorem on elimination of points of indeterminacy.

Geometry of Robinson consistency in Łukasiewicz logic

We establish the Robinson joint consistency theorem for the infinite-valued propositional logic of Łukasiewicz. As a corollary we easily obtain the amalgamation property for MV-algebras—the algebras of Łukasiewicz logic: all pre-existing proofs of this latter result make essential use of the Pierce amalgamation theorem for abelian lattice-ordered groups (with strong unit) together with the categorical equivalence Γ between these groups and MV-algebras. Our main tools are elementary and geometric.

Free generating sets of lattice-ordered abelian groups

We prove that the generators g1,…,gn of a lattice-ordered abelian group G form a free generating set iff each ℓ-ideal generated by any n−1 linear combinations of the gi is strictly contained in some maximal ℓ-ideal of G.

The underlying group of any finitely generated Abelian lattice-ordered group is free

The underlying group of any finitely generated Abelian lattice-ordered group is free

The Lebesgue state of a unital abelian lattice-ordered group

The Lebesgue state of a unital abelian lattice-ordered group

Lattice-ordered Abelian groups and Schauder bases of unimodular fans

Baker-Beynon duality theory yields a concrete representation of any finitely generated projective Abelian lattice-ordered group G G in terms of piecewise linear homogeneous functions with integer coefficients, defined over the support | Σ | |\Sigma | of a fan Σ \Sigma . A unimodular fan Δ \Delta over | Σ | |\Sigma | determines a Schauder basis of G G : its elements are the minimal positive free generators of the pointwise ordered group of Δ \Delta -linear support functions. Conversely, a Schauder basis H \mathbf {H} of G G determines a unimodular fan over | Σ | |\Sigma | : its maximal cones are the domains of linearity of the elements of H \mathbf {H} . The main purpose of this paper is to give various representation-free characterisations of Schauder bases. The latter, jointly with the De Concini-Procesi starring technique, will be used to give novel characterisations of finitely generated projective Abelian lattice ordered groups. For instance, G G is finitely generated projective iff it can be presented by a purely lattice-theoretical word.

On the variety generated by all nilpotent lattice-ordered groups

In 1974, J. Martinez introduced the variety ${\mathcal W}$ of weakly Abelian lattice-ordered groups; it is defined by the identity \[ x^{-1}(y\vee 1)x\vee (y\vee 1)^2=(y\vee 1)^2.\]

Bernoulli automorphisms of finitely generated free MV-algebras

MV-algebras can be viewed either as the Lindenbaum algebras of Łukasiewicz infinite-valued logic, or as unit intervals [ 0 , u ] of lattice-ordered abelian groups in which a strong order unit u > 0 has been fixed. They form an equational class, and the free n -generated free MV-algebra is representable as an algebra of piecewise-linear continuous functions with integer coefficients over the unit n -dimensional cube. In this paper we show that the automorphism group of such a free algebra contains elements having strongly chaotic behaviour, in the sense that their duals are measure-theoretically isomorphic to a Bernoulli shift. This fact is noteworthy from the viewpoint of algebraic logic, since it gives a distinguished status to Lebesgue measure as an averaging measure on the space of valuations. As an ergodic theory fact, it provides explicit examples of volume-preserving homeomorphisms of the unit cube which are piecewise-linear with integer coefficients, preserve the denominators of rational points, and enjoy the Bernoulli property.

Elementary Theory for l-Ideal Lattices of Abelian l-Groups

A description is furnished for ideal lattices of free vector lattices \(F\) with a finite number n of free generators. It is proved that an elementary theory for l-ideal lattices of Abelian lattice-ordered groups is undecidable, which gives a negative solution to Problem 5.20 in the “Kourovka Notebook.”