Countable models of ℵ1-categorical theories

Countable models of ℵ₁-categorical theories are classified. It is shown that such a theory has only a countable number of nonisomorphic countable models. The author was partially supported by NSF grants GP-1621 and GP-4257 during the period those results were obtained.     

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.     

The countable existentially closed pseudocomplemented semilattice

As the class \(\mathcal {PCSL}\) of pseudocomplemented semilattices is a universal Horn class generated by a single finite structure it has a \(\aleph _0\)-categorical model companion \(\mathcal {PCSL}^*\). As \(\mathcal {PCSL}\) is inductive the models of \(\mathcal {PCSL}^*\) are exactly the existentially closed models of \(\mathcal {PCSL}\). We will construct the unique existentially closed countable model of \(\mathcal {PCSL}\) as a direct limit of algebraically closed pseudocomplemented semilattices.     

On existentially closed and generic nilpotent groups

Letn≧2 be an integer. We prove the following results that are known in casen=2: The upper and the lower central series of an existentially closed nilpotent group of classn coincide. A finitely generic nilpotent group of classn is periodic and the center of a finitely generic torsion-free nilpotent group of classn is isomorphic toQ ⁺, whereas infinitely generic nilpotent groups do not enjoy these properties. We determine the structure of the torsion subgroup of existentially closed nilpotent groups of class 2. Finally we give an algebraic proof that there exist 2^κ non-isomorphic existentially closed nilpotent groups of classn in cardinalityK ≧N ₀. Some results of this paper were contained in [6].     

Simple,existentially closed extensions of unoids

Translated from Matematicheskie Zametki, Vol. 44, No. 4, pp. 449–456, October, 1988.     

Finite axiomatizations for existentially closed posets and semilattices

In this paper we exhibit axiomatizations for the theories of existentially closed posets and existentially closed semilattices. We do this by considering an infinite axiomatization which characterizes these structures in terms of embeddings of finite substructures, an axiomatization which exists for any locally finite universal class with a finite language and with the joint embedding and amalgamation properties. We then find particular finite subsets of these axioms which suffice to axiomatize both classes. Research supported by an NSERC Postdoctoral Fellowship. Research supported by NSERC Grant No. A7256.     

The class of algebraically closed p-semilattices is finitely axiomatizable

We prove our title, and thereby establish the base for a positive solution of Albert and Burris’ problem on the finite axiomatizability of the model companion of the class of all pseudocomplemented semilattices.     

The theory of integrally closed domains is not finitely axiomatizable

It is well‐known that the theory of algebraically closed fields is not finitely axiomatizable. In this note, we prove that the theory of integrally closed integral domains is also not finitely axiomatizable.     

Theories with only a finite number of existentially complete models

We construct, in particular, a countable universal theory with JEP which has exactly 2 non-isomorphic countable existentially complete models, and these two models can be either elementarily equivalent or inequivalent.