Torsion classes of generalized Boolean algebras

Torsion classes and radical classes of lattice ordered groups have been investigated in several papers. The notions of torsion class and of radical class of generalized Boolean algebras are defined analogously. We denote by T _g and R _g the collections of all torsion classes or of all radical classes of generalized Boolean algebras, respectively. Both T _g and R _g are partially ordered by the class-theoretical inclusion. We deal with the relation between these partially ordered collection; as a consequence, we obtain that T _g is a Brouwerian lattice. W. C. Holland proved that each variety of lattice ordered groups is a torsion class. We show that an analogous result is valid for generalized Boolean algebras.     

k-metric spaces

In this paper, we give a new generalization of metric spaces called k-metric spaces. Our k-metrics are valued in lattice ordered groups, which allows us to talk about distance in non-abelian lattice ordered groups. We also discuss a class of (not necessarily abelian) lattice ordered groups in which every k-metric induces a topology. Then we show that every k-metric valued in the real numbers is metrizable. In the last section, we characterize intrinsic metrics on lattice ordered rings that are almost f-rings and prove that being an almost f-ring is necessary and sufficient for this characterization. Then we show that if a lattice ordered ring is representable, then every intrinsic metric is a k-metric.     

On a type of distributivity of lattice ordered groups

Let m be an infinite cardinal. Inspired by a result of Sikorski on m-representability of Boolean algebras, we introduce the notion of r _m-distributive lattice ordered group. We prove that the collection of all such lattice ordered groups is a radical class. Using the mentioned notion, we define and investigate a homogeneity condition for lattice ordered groups.     

Topologies on abelian lattice ordered groups induced by a positive filter and completeness

We consider topologies on an abelian lattice ordered group that are determined by the absolute value and a positive filter. We show that the topological completions of these objects are also determined by the absolute value and a positive filter. We investigate the connection between the topological completion of such objects and the Dedekind–MacNeille completion of the underlying lattice ordered group. We consider the preservation of completeness for such topologies with respect to homomorphisms of lattice ordered groups. Finally, we show that topologies defined in terms of absolute value and a positive filter on the space C(X) of all real-valued continuous functions defined on a completely regular topological space X are always complete.     

Some theorems on graphs and posets

In this journal, Leclerc proved that the dimension of the partially ordered set consisting of all subtrees of a tree T, ordered by inclusion, is the number of end points of T. Leclerc posed the problem of determining the dimension of the partially ordered set P consisting of all induced connected subgraphs of a connected graph G for which P is a lattice.In this paper, we prove that the poset P consisting of all induced connected subgraphs of a nontrivial connected graph G, partially ordered by inclusion, has dimension n where n is the number of noncut vertices in G whether or not P is a lattice. We also determine the dimension of the distributive lattice of all subgraphs of a graph.     

A proof of Frankl’s union-closed sets conjecture for dismantlable lattices

For a lattice L, let Princ(L) denote the ordered set of principal congruences of L. In a pioneering paper, G. Grätzer characterized the ordered set Princ(L) of a finite lattice L; here we do the same for a countable lattice. He also showed that every bounded ordered set H is isomorphic to Princ(L) of a bounded lattice L. We prove a related statement: if an ordered set H with a least element is the union of a chain of principal ideals (equivalently, if 0 \({\in}\) H and H has a cofinal chain), then H is isomorphic to Princ(L) of some lattice L.     

Definability in substructure orderings, III: Finite distributive lattices

Let ${{\mathcal D}}$ be the ordered set of isomorphism types of finite distributive lattices, where the ordering is by embeddability. We study first-order definability in this ordered set. We prove among other things that for every finite distributive lattice D, the set {d, d ^opp} is definable, where d and d ^opp are the isomorphism types of D and its opposite (D turned upside down). We prove that the only non-identity automorphism of ${{\mathcal D}}$ is the opposite map. Then we apply these results to investigate definability in the closely related lattice of universal classes of distributive lattices. We prove that this lattice has only one non-identity automorphism, the opposite map; that the set of finitely generated and also the set of finitely axiomatizable universal classes are definable subsets of the lattice; and that for each element K of the two subsets, {K, K ^opp} is a definable subset of the lattice.     

Intrinsic metrics for lattice ordered groups

Swamy and Jakubik studied the metric ¦x y¦ on lattice ordered groups, and isometries which presere it. We show the only intrinsic metrics on lattice ordered groups are the multiplesn ¦x–y ¦ of theirs, and that the triangle inequality is satisfied by such a metric iff the group is abelian. We show that there are isometries for each of these metrics, but they are rare. We give a simpler proof via permutation groups of the following augmented version of a theorem of Jakubik. IfT is an isometry of the lattice ordered groupG with respect to the metric ¦x¥¦ andT(0)=0, thenG=AB, B is abelian, andT(a+b)=a–b; conversely, any suchT is an isometry.To Paul Conrad on his 60th birthdayPresented by L. Fuchs.     

Ordered semigroups which are both right commutative and right cancellative

In this paper we prove that each right commutative, right cancellative ordered semigroup (S,.,??) can be embedded into a right cancellative ordered semigroup (T,??,?) such that (T,??) is left simple and right commutative. As a consequence, an ordered semigroup S which is both right commutative and right cancellative is embedded into an ordered semigroup T which is union of pairwise disjoint abelian groups, indexed by a left zero subsemigroup of?T.     

Definibility in normal theories

This paper initiates an investigation which seeks to explain elementary definability as the classical results of mathematicallogic (the completeness, compactness and Löwenheim-Skolem theorems) explain elementary logical consequence. The theorems of Beth and Svenonius are basic in this approach and introduce automorphism groups as a means of studying these problems. It is shown that for a complete theoryT, the definability relation of Beth (or Svenonius) yields an upper semi-lattice whose elements (concepts) are interdefinable formulas ofT (formulas having equal automorphism groups in all models ofT). It is shown that there are countable modelsA ofT such that two formulae are distinct (not interdefinable) inT if and only if they are distinct (have different automorphism groups) inA. The notion of a concepth being normal in a theoryT is introduced. Here the upper semi-lattice of all concepts which defineh is proved to be a finite lattice—anti-isomorphic to the lattice of subgroups of the corresponding automorphism group. Connections with the Galois theory of fields are discussed.     

Fractional parts of dense additive subgroups of real numbers

Given a dense additive subgroup G of \(\mathbb {R}\) containing \(\mathbb {Z}\), we consider its intersection \(\mathbb {G}\) with the interval [0, 1[ with the induced order and the group structure given by addition modulo 1. We axiomatize the theory of \(\mathbb {G}\) and show it is model-complete, using a Feferman-Vaught type argument. We show that any sufficiently saturated model decomposes into a product of a standard part and two ordered semigroups of infinitely small and infinitely large elements.     

Representing some families of monotone maps by principal lattice congruences

For a lattice L with 0 and 1, let Princ(L) denote the set of principal congruences of L. Ordered by set inclusion, it is a bounded ordered set. In 2013, G. Grätzer proved that every bounded ordered set is representable as Princ(L); in fact, he constructed L as a lattice of length 5. For {0, 1}-sublattices \({A \subseteq B}\) of L, congruence generation defines a natural map Princ(A) \({\longrightarrow}\) Princ(B). In this way, every family of {0, 1}-sublattices of L yields a small category of bounded ordered sets as objects and certain 0-separating {0, 1}-preserving monotone maps as morphisms such that every hom-set consists of at most one morphism. We prove the converse: every small category of bounded ordered sets with these properties is representable by principal congruences of selfdual lattices of length 5 in the above sense. As a corollary, we can construct a selfdual lattice L in G. Grätzer's above-mentioned result.     

Coverings of [MO n ] and minimal orthomodular lattices

If T is an orthomodular lattice (OML), we denote by [T] the equational class generated by T. In this paper we characterize the finite OMLs T such that [T] covers some [MO _n ]. These OMLs T are the non-modular OMLs such that all proper sub-OMLs of T are modular. An OML satisfying that last property is called minimal. There exist infinitely many minimal OMLs provided by quadratic spaces over finite fields. We describe them and give a new way to represent their Greechie diagrams in two separate parts. Other methods to obtain finite minimal OMLs are given. Received May 14, 2005; accepted in final form May 30, 2007.     

Functorial topologies and finite-index subgroups of abelian groups

In the general context of functorial topologies, we prove that in the lattice of all group topologies on an abelian group, the infimum between the Bohr topology and the natural topology is the profinite topology. The profinite topology and its connection to other functorial topologies is the main objective of the paper. We are particularly interested in the poset C(G) of all finite-index subgroups of an abelian group G, since it is a local base for the profinite topology of G. We describe various features of the poset C(G) (its cardinality, its cofinality, etc.) and we characterize the abelian groups G for which C(G)?{G} is cofinal in the poset of all subgroups of G ordered by inclusion. Finally, for pairs of functorial topologies T, S we define the equalizer E(T,S), which permits to describe relevant classes of abelian groups in terms of functorial topologies.     

Invariant subsets of scattered trees and the tree alternative property of Bonato and Tardif

A tree is scattered if it does not contain a subdivision of the complete binary tree as a subtree. We show that every scattered tree contains a vertex, an edge, or a set of at most two ends preserved by every embedding of T. This extends results of Halin, Polat and Sabidussi. Calling two trees equimorphic if each embeds in the other, we then prove that either every tree that is equimorphic to a scattered tree T is isomorphic to T, or there are infinitely many pairwise non-isomorphic trees which are equimorphic to T. This proves the tree alternative conjecture of Bonato and Tardif for scattered trees, and a conjecture of Tyomkyn for locally finite scattered trees.     

The Lattice of Subvarieties of Semilattice Ordered Algebras

This paper is devoted to the semilattice ordered \(\mathcal{V}\) -algebras of the form (A, Ω, +?), where + is a join-semilattice operation and (A, Ω) is an algebra from some given variety \(\mathcal{V}\) . We characterize the free semilattice ordered algebras using the concept of extended power algebras. Next we apply the result to describe the lattice of subvarieties of the variety of semilattice ordered \(\mathcal{V}\) -algebras in relation to the lattice of subvarieties of the variety \(\mathcal{V}\) .     

On the multiplicity of homoclinic orbits on Riemannian manifolds for a class of second order Hamiltonian systems

We prove the existence of infinitely many homoclinic orbits on a Riemannian manifold (possibly non-compact), for a class of second order Hamiltonian systems of the form: $$D_t \dot x(t) + grad_x V(t,x(t)) = 0$$ where the potentialV isT-periodic in the time variable.     

Definability in Substructure Orderings,II: Finite Ordered Sets

Let \({{\uppercase {\mathcal{p}}}} \) be the ordered set of isomorphism types of finite ordered sets (posets), where the ordering is by embeddability. We study first-order definability in this ordered set. We prove among other things that for every finite poset P, the set \(\{p,p^{\partial}\}\) is definable, where p and \(p^{\partial}\) are the isomorphism types of P and its dual poset. We prove that the only non-identity automorphism of \({{\uppercase {\mathcal{p}}}}\) is the duality map. Then we apply these results to investigate definability in the closely related lattice of universal classes of posets. We prove that this lattice has only one non-identity automorphism, the duality map; that the set of finitely generated and also the set of finitely axiomatizable universal classes are definable subsets of the lattice; and that for each member K of either of these two definable subsets, \(\{K,K^{\partial}\}\) is a definable subset of the lattice. Next, making fuller use of the techniques developed to establish these results, we go on to show that every isomorphism-invariant relation between finite posets that is definable in a certain strongly enriched second-order language \(\textup{\emph L}_2\) is, after factoring by isomorphism, first-order definable up to duality in the ordered set \({{\uppercase {\mathcal{p}}}}\). The language \(\textup{\emph L}_2\) has different types of quantifiable variables that range, respectively, over finite posets, their elements and order-relation, and over arbitrary subsets of posets, functions between two posets, subsets of products of finitely many posets (heteregenous relations), and can make reference to order relations between elements, the application of a function to an element, and the membership of a tuple of elements in a relation.     

Tensor envelopes of regular categories

We extend the calculus of relations to embed a regular category A into a family of pseudo-abelian tensor categories T(A,δ) depending on a degree function δ. Assume that all objects have only finitely many subobjects. Then our results are as follows:

1.

Let N be the maximal proper tensor ideal of T(A,δ). We show that T(A,δ)/N is semisimple provided that A is exact and Mal'cev. Thereby, we produce many new semisimple, hence abelian, tensor categories.

2.

Using lattice theory, we give a simple numerical criterion for the vanishing of N.

3.

We determine all degree functions for which T(A,δ)/N is Tannakian. As a result, we are able to interpolate the representation categories of many series of profinite groups such as the symmetric groups Sn, the hyperoctahedral groups , or the general linear groups GL(n,Fq) over a fixed finite field.

This paper generalizes work of Deligne, who first constructed the interpolating category for the symmetric groups Sn.     

Higher rank subgroups in the class groups of imaginary function fields

Let F be a finite field and T a transcendental element over F. In this paper, we construct, for integers m and n relatively prime to the characteristic of F(T), infinitely many imaginary function fields K of degree m over F(T) whose class groups contain subgroups isomorphic to (Z/nZ)^m. This increases the previous rank of m−1 found by the authors in [Y. Lee, A. Pacelli, Class groups of imaginary function fields: The inert case, Proc. Amer. Math. Soc. 133 (2005) 2883-2889].