The word problem in free normal valued lattice-ordered groups: a solution and practical shortcuts

In any lattice-ordered group (l-group) generated by a setX, every element can be written (not uniquely) in the form w(x)=⋁ _i ⋀ _j w _ij (x), where eachw _ij (x) is a group word in the elements ofX. An algorithm will be given for deciding whetherw(x) is the identitye in the free normal valuedl-group onX, or equivalently, whether the statement “∀x,w(x)=e” holds in all normal valuedl-groups. The algorithm is quite different from the one given recently by Holland and McCleary for the freel-group, and indeed the solvability of the word problem was established first for the normal valued case. The present algorithm makes crucial use of the fact (due to Glass, Holland, and McCleary) that the variety of normal valuedl-groups is generated by the finite wreath powersZ Wr Z Wr...Wr Z of the integersZ. In general, use of the algorithm requires a fairly large amount of work, but in several important special cases shortcuts are obtained which make the algorithm very quick. This is an expanded version of material developed while the author was on leave at Bowling Green State University in Bowling Green, Ohio, and presented in 1978 at the Conference on Ordered Groups at Boise State University in Boise, Idaho [9]. Presented by L. Fuchs.     

Special-valuedl-groups

Special elements and special values have always been of interest in the study of lattice-ordered groups, arising naturally from totally-ordered groups and lexicographic extensions. Much work has been done recently with the class of lattice-ordered groups whose root system of regular subgroups has a plenary subset of special values. We call suchl-groupsspecial- valued. In this paper, we first show that several familiar structures, namely polars, minimal prime subgroups, and the lex kernel, are recognizable from the lattice and the identity. This then leads to an easy proof that special elements can also be recognized from the lattice and the identity. We then give a simple and direct proof thatl, the class of special-valuedl-groups, is closed with respect to joins of convexl-subgroups, incidentally giving a direct proof thatl is a quasitorsion class. This proof is then used to show that the special-valued and finite-valued kernels ofl-groups are recognizable from the lattice and the identity. We also show that the lateral completion of a special-valuedl-group is special-valued and is an a^*-extension of the originall-group.Our most important result is that the lateral completion of a completely distributive normal-valuedl-group is special-valued. This lends itself easily to a new and simple proof of a result by Ball, Conrad, and Darnel that generalizes the Conrad-Harvey-Holland Theorem, namely, that every normal-valuedl-group can be -embedded into a special-valuedl-group.This paper is dedicated to the memory of Prof. Samuel Wolfenstein, who initiated the study of normal-valuedl-groups and recognized early the importance of special-valuedl-groups.Presented by L. Fuchs.     

Uniqueness of the group operation on the lattice of order-automorphisms of the real line

A(R) is the lattice-ordered group (l-group) of all order-automorphisms of the real lineR, with the usual pointwise order and “of course” with composition as the group operation. In fact, what other choices are there for a group operation having the same identity that would give anl-group? Composition in the reverse order would work. But there are no other choices — the group operation can be recognized in the lattice. Several classes of abelianl-groups having a unique group operation have been found by Conrad and Darnel, but this is the first non-abelian example having the minimum of two group operations. “Conversely”, Holland has shown that for the groupA(R) under composition, the only lattice orderings yielding anl-group are the pointwise order and its dual. These results also hold for the rational lineQ.     

Some remarks on almost l-groups

The divisibility group of every Bézout domain is an abelian l-group. Conversely, Jaffard, Kaplansky, and Ohm proved that each abelian l-group can be obtained in this way, which generalizes Krull’s theorem for abelian linearly ordered groups. Dumitrescu, Lequain, Mott, and Zafrullah [3] proved that an integral domain is almost GCD if and only if its divisibility group is an almost l-group. Then they asked whether the Krull-Jaffard-Kaplansky-Ohm theorem on l-groups can be extended to the framework of almost l-groups, and asked under what conditions an almost l-group is lattice-ordered [3, Questions 1 and 2]. This note answers the two questions. Received: 29 April 2008     

Distinguished extensions of a lattice-ordered group

Any lattice-ordered group (l-group for short) is essentially extended by its lexicographic product with a totally ordered group. That is, anl-homomorphism (i.e., a group and lattice homomorphism) on the extension which is injective on thel-group must be injective on the extension as well. Thus nol-group has a maximal essential extension in the categoryIGp ofl-groups withl-homomorphisms. However, anl-group is a distributive lattice, and so has a maximal essential extension in the categoryD of distributive lattices with lattice homomorphisms. Adistinguished extension of onel-group by another is one which is essential inD. We characterize such extensions, and show that everyl-groupG has a maximal distinguished extensionE(G) which is unique up to anl-isomorphism overG.E(G) contains most other known completions in whichG is order dense, and has mostl-group completeness properties as a result. Finally, we show that ifG is projectable then E(G) is the -completion of the projectable hull ofG.Presented by M. Henriksen.     

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, Nullstellensätze 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.     

Groups of Automorphisms of Tournaments

By the Holland Representation Theorem, every lattice ordered group (l-group) is isomorphic to a subalgebra of the l-group of automorphisms of a chain. Since weakly associative lattice groups (wal-groups) and tournaments are non-transitive generalizations of l-groups and chains, respectively, the problem concerning the possibility of representation of wal-groups by automorphisms of tournaments arises. In the paper we describe the class of wal-groups isomorphic to wal-groups of automorphisms of tournament and show some of its properties.     

Signatures andS-discrete lattice-ordered groups

The central theme of this article is the approximation of lattice-ordered groups (l-groups) first by Specker groups and, subsequently, by the so-calledS-discretel-groups. The sense of these approximations is made precise via the notion of a signature, defined below, and by that ofa ^*-subgroups. Sample result: ifG is a projectablel-group then it has anl-subgroupH which is Specker and for which the mapPPH defines a boolean isomorphism between the algebras of polars ofG andH.Presented by L. Fuchs.This article was written while this author was a Stouffer Visiting Professor at the University of Kansas. He wishes to thank the members of the Mathematics Department of that institution for their hospitality.     

Minimal Number of Generators and Minimum Order of a Non-Abelian Group Whose Elements Commute with Their Endomorphic Images

A group in which every element commutes with its endomorphic images is called an “E-group″. If p is a prime number, a p-group G which is an E-group is called a “pE-group″. Every abelian group is obviously an E-group. We prove that every 2-generator E-group is abelian and that all 3-generator E-groups are nilpotent of class at most 2. It is also proved that every infinite 3-generator E-group is abelian. We conjecture that every finite 3-generator E-group should be abelian. Moreover, we show that the minimum order of a non-abelian pE-group is p ⁸ for any odd prime number p and this order is 2⁷ for p = 2. Some of these results are proved for a class wider than the class of E-groups.     

Projectable and strongly projectable lattice-ordered groups

This paper is concerned with the classes and of weakly projectable, projectable, and strongly projectablel-groups (lattice-ordered groups). It is shown that the members of ] and can be characterized purely in terms of their order structures, and these characterizations are used in establishing, among other things, that lattice isomorphisms preserve projectability and strong projectability. Further characterizations in terms of the lattice of convexl-subgroups are also given. Additional results include the following: The existence of an indecomposable, weakly projectable archimedeanl-group; the fact that the -radical of a laterally completel-group is a conditionally orthocompletel-group; and finally, the result that the -radical of anl-groupG contains every strongly projectablel-group that is large inG. Presented by R. McKenzie.In memoriam Jürgen Schmidt     

Lamron ℓ-groups

The article introduces a new class of lattice-ordered groups. An ?-group G is lamron if Min(G)^?1 is a Hausdorff topological space, where Min(G)^?1 is the space of all minimal prime subgroups of G endowed with the inverse topology. It will be evident that lamron ?-groups are related to ?-groups with stranded primes. In particular, it is shown that for a W-object (G,u), if every value of u contains a unique minimal prime subgroup, then G is a lamron ?-group; such a W-object will be said to have W-stranded primes. A diverse set of examples will be provided in order to distinguish between the notions of lamron, stranded primes, W-stranded primes, complemented, and weakly complemented ?-groups.     

Partial Decomposition Bases and Global Warfield Groups

The class of abelian groups with partial decomposition bases was developed by the first author in order to generalize Barwise and Eklof's classification of torsion groups in L_∞ω. In this article, we continue to explore algebraic characteristics of this class and establish a uniqueness theorem, extending our previous work on mixed p-local groups to the global case. It is shown that groups with partial decomposition bases are characterized in terms of Warfield groups and k-groups of Hill and Megibben. In fact, we prove that the class of groups with partial decomposition bases is identical to the class of k-groups, and, as such, closed under direct summands, and that every finitely generated subgroup of a k-group is locally nice. Also, we introduce and explore subgroups possessing a partial subbasis. As an application, it is shown that isotype k-subgroups of abelian groups are k-groups.     

Normal subgroups and elementary theories of lattice-ordered groups

We show that any lattice-ordered group (l-group) G can be l-embedded into continuously many l-groups H _i which are pairwise elementarily inequivalent both as groups and as lattices with constant e. Our groups H _i can be distinguished by group-theoretical first-order properties which are induced by lattice-theoretically nice properties of their normal subgroup lattices. Moreover, they can be taken to be 2-transitive automorphism groups A(S _i ) of infinite linearly ordered sets (S _i , ) such that each group A(S _i ) has only inner automorphisms. We also show that any countable l-group G can be l-embedded into a countable l-group H whose normal subgroup lattice is isomorphic to the lattice of all ideals of the countable dense Boolean algebra B.     

Solvable andl-solvablel-groups

It is possible to consider the question of solvability of a lattice-ordered group via two different approaches — one involves the satisfying of appropriate commutator laws by thel-group and the other involves the study of the derived series ofl-ideals of thel-group. The first method defines solvability strictly in terms of group operations, while the second deals with the lattice-ordering as well; hence, solvability andl-solvability. This paper deals with some of the distinctions between these two families ofl-group varieties.Presented by L. Fuchs.     

Varieties minimal over representable varieties of lattice-ordered groups

Weinberg showed that the variety of abelian lattice-ordered groups is the minimal nontrivial variety in the lattice of varieties of lattice-ordered groups. Scrimger showed that the abelian variety of lattice-ordered groups has countably infinitely many nonrepresentable covering varieties, and it is now known that his varieties are the only nonrepresentable covers of the abelian variety.

In this paper, a variation of the method used to construct the Scrimger varieties is developed that is shown to produce every nonrepresentable cover of any representable variety. Using this variation, all nonrepresentable covers of any weakly abelian l-variety are specifically identified, as are the nonrepresentable covers of any l-metabelian representable l-variety. In both instances, such il-varieties have only countably infinitely many such covers.

Any nonrepresentable cover of a representable il-variety is shown to be a subvariety of a quasi-representable il-variety as defined by Reilly. The class of these quasi-representable l-varieties is shown to contain the well-known L_n l-varieties and to generalize many of their properties.     

Groupable lattices

A lattice is called groupable provided it can be endowed with the structure of an l-group (lattice ordered group). The primary objective of this paper is to introduce an order theoretic property of groupable lattices which implies that all associated l-groups are subdirect products of totally ordered groups. This is an analog to Iwasawa's well-known result which asserts that a conditionally complete l-group is abelian. A secondary objective is to outline a general method for identifying classes of l-groups determined by order theoretic properties.     

On groups with the almost orthogonality condition

The concept of the orthogonality is important in the study of lattice-ordered groups. The purpose of this paper is to investigate a generalization of this concept to directed groups. The idea of an AO-group is introduced here. Some results are obtained concerning convex directed subgroups and Archimedean extensions of AO-groups. Interpolation AO-groups are also described.     

On archimedean extensions ofAO-groups

The concept of almost orthogonality is a variant for extending the concept of the orthogonal elements of lattice-ordered groups to arbitrary partially ordered groups. In the present paper, the notion of an Archimedean extension of anAO-groups is studied. Some results are obtained concerning interpolationAO-groups. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 74, Algebra-15, 2000.     

Finite groups with some pronormal subgroups

A subgroup H of finite group G is called pronormal in G if for every element x of G, H is conjugate to H ^x in 〈H, H ^x 〉. A finite group G is called PRN-group if every cyclic subgroup of G of prime order or order 4 is pronormal in G. In this paper, we find all PRN-groups and classify minimal non-PRN-groups (non-PRN-group all of whose proper subgroups are PRN-groups). At the end of the paper, we also classify the finite group G, all of whose second maximal subgroups are PRN-groups.     

Quasi-varieties of lattice ordered groups

Whereas there is a maximal proper variety of lattice-ordered groups, it is known that there is no maximal proper quasi-variety of lattice-ordered groups. We prove that there are 2º (the maximal possible) pairwise incomparable quasi-varieties of lattice-ordered groups containing . Some of the distributive laws of the semigroup lattice of quasi-varieties are examined and their truth (or falsity) is established. It is also shown here that the latticeL of alll-group varieties is a sublattice of the latticeQ of quasi-varieties ofl-groups but fails to be a complete sublattice.This article is a part of the author's Ph.D. dissertation which was directed by Professor A. M. W. Glass. The author wishes to express his sincere gratitude to Professor Glass for his assistance and encouragement during the writing of the dissertation and this article. He also wishes to thank Professor K. K. Hickin for his help with nilpotent material, in particular for his help in establishing Theorem 4.7.Presented by L. Fuchs.