Anti-elementarity is a strong way of ensuring that a class of structures , in a given first-order language, is not closed under elementary equivalence with respect to any infinitary language of the form L ∞λ. We prove that many naturally defined classes are anti-elementary, including the following: • the class of all lattices of finitely generated ...
Published in Mathematica Slovaca
In the category Arch of archimedean l-groups, the r.u. completion of the divisible hull, rdA, is the maximum essential reflection and the maximum majorizing reflection (Ball-Hager, 1999). In the weak-unital subcategory W, the reflections c3A and mA (Aron-Hager, 1981) are respectively maximum essential, and maximum majorizing (Ball-Hager, 1993), and...
Published in Mathematica Slovaca
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.
Published in Mathematica Slovaca
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 copi...
Published in Algebra universalis
In the category of archimedean lattice-ordered groups with strong unit, the Yosida representation provides an insightful description of coproducts and the condition that an object be hyperarchimedean. These are combined to yield a criterion that a coproduct be hyperarchimedean and this is applied, viz.: G∐G\documentclass[12pt]{minimal} \usepackage{...
Published in Mathematica Slovaca
A ring with identity is said to be clean if every element can be written as a sum of a unit and an idempotent. The study of clean rings has been at the forefront of ring theory over the past decade. The theory of partially-ordered groups has a nice and long history and since there are several ways of relating a ring to a (unital) partially-ordered ...
Published in Algebra universalis
Within the lattice of varieties of pseudo MV-algebras, the variety \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{B}}$$\end{document} of Boolean algebras is ...
Published in Milan Journal of Mathematics
In a recent paper, F. Boca investigates the AF algebra \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathfrak{A}}}$$\end{document} associated with the Farey-Stern-...
Published in Mathematica Slovaca
This issue of Mathematica Slovaca is in honour of W. Charles Holland’s 75th birthday. We present here a brief account of some of his research (to date) and a couple of brief personal sketches of the man.
Published in Mathematica Slovaca
Let H i be a sublattice subgroup of a lattice-ordered group G i (i = 1, 2). Suppose that H 1 and H 2 are isomorphic as lattice-ordered groups, say by φ. In general, there is no lattice-ordered group in which G 1 and G 2 can be embedded (as lattice-ordered groups) so that the embeddings agree on the images of H 1 and H 1φ. In this article we prove t...
