On the variety generated by all nilpotent lattice-ordered groups

In 1974, J. Martinez introduced the variety of weakly Abelian lattice-ordered groups; it is defined by the identity

     

Shape resonances in slow electron scattering by aromatic molecules. I. Anthraquinone derivatives

A series of anthraquinone (C(14)O(2)H(8)) derivatives has been studied by means of electron capture negative ion mass spectrometry (ECNI-MS), photoelectron spectroscopy (PES), and AM1 quantum chemical calculations. Mean lifetimes of molecular negative ions M(-.) (MNI) have been measured. The mechanism of long-lived MNI formation in the epithermal energy region of incident electrons has been investigated. A simple model of a molecule (a spherical potential well with the repulsive centrifugal term) has been applied for the analysis of the energy dependence of cross sections at the first stage of the electron capture process. It has been shown that a temporary resonance of MNI at the energy approximately 0.5 eV corresponds to a shape resonance with lifetime 1-2.10(-13) s in the f-partial wave (l = 3) of the incident electron. The next resonant state of MNI at the energy approximately 1.7 eV has been associated with the electron excited Feshbach resonance (whose parent state is a triplet npi* transition). In all cases the initial electron state of the MNI relaxes into the ground state by means of a radiationless transition, and the final state of the MNI is a nuclear excited resonance with a lifetime measurable on the mass spectrometry timescale. Copyright 1999 John Wiley & Sons, Ltd.     

Amalgamation bases for the class of lattice-ordered groups

We prove Theorem A. 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 Theorem B. 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.     

Completion of Linearly Ordered Groups

We give examples of linearly ordered groups that are not embeddable in divisible orderable. In the first example, the group does not embed in any divisible group with strictly isolated unity. In the second example, the group in question is an O*-group, and in the third, it is a group with a central system of convex subgroups. To my teacher A. I. Kokorin Supported by RFBR grant Nos. 96-01-00358, 99-01-00335, and 03-01-00320. __________ Translated from Algebra i Logika, Vol. 44, No. 6, pp. 664–681, November–December, 2005.     

Normal relatively convex subgroups of solvable orderable groups

Orderable solvable groups in which every relatively convex subgroup is normal are studied. If such a class is subgroup closed than it is precisely the class of solvable orderable groups which are locally of finite (Mal’tsev) rank. A criterion for an orderable metabelian group to have every relatively convex subgroup normal is given. Examples of an orderable solvable group G of length three with periodic G/G′ and of an orderable solvable group of length four with only one proper normal relatively convex subgroup are constructed. To the memory of N. Ya. Medvedev Supported by RFBR (project No. 03-01-00320). Translated from Algebra i Logika, Vol. 48, No. 3, pp. 291–308, May–June, 2009.     

Automorphism groups of totally ordered sets: A retrospective survey

In 1963, W. Charles Holland proved that every lattice-ordered group can be embedded in the lattice-ordered group of all order-preserving permutations of a totally ordered set. In this article we examine the context and proof of this result and survey some of the many consequences of the ideas involved in this important theorem.