On Vaught’s Conjecture and finitely valued MV algebras

We show that the complete first order theory of an MV algebra has $2^{\aleph _0}$ countable models unless the MV algebra is finitely valued. So, Vaught's Conjecture holds for all MV algebras except, possibly, for finitely valued ones. Additionally, we show that the complete theories of finitely valued MV algebras are $2^{\aleph _0}$ and that all ω‐categorical complete theories of MV algebras are finitely axiomatizable and decidable. As a final result we prove that the free algebra on countably many generators of any locally finite variety of MV algebras is ω‐categorical.     

Automorphism groups of decidable models

It is proved that there exists no relationship between isomorphism types of the ordinary and recursive automorphism groups of recursive models and the property of being decidable for these models. Moreover, we show that all isomorphism types of recursive automorphism groups can be realized in a single (up to isomorphism) decidable countably categorical model. Taking account of the action of a group on the universe of the model makes it possible to distinguish between the classes of groups for decidable and all the recursive models.Translated fromAlgebra i Logika, Vol. 34, No. 4, pp. 437–447, July-August, 1995.     

Complexity of Categorical Theories with Computable Models

M. Lerman and J. Scmerl specified some sufficient conditions for computable models of countably categorical arithmetical theories to exist. More precisely, it was shown that if T is a countably categorical arithmetical theory, and the set of its sentences beginning with an existential quantifier and having at most n+1 alternations of quantifiers is _n+1 ⁰ for any n, then T has a computable model. J. Night improved this result by allowing certain uniformity and omitting the requirement that T is arithmetical. However, all of the known examples of theories of₀-categorical computable models had low level of algorithmic complexity, and whether there are theories that would satisfy the above conditions for sufficiently large n was unknown. This paper will include such examples.     

Fraïssé’s Construction from a Topos-Theoretic Perspective

We present a topos-theoretic interpretation of (a categorical generalization of) Fraïssé’s construction in Model Theory, with applications to homogeneous models and countably categorical theories.     

Complete Theories with Finitely Many Countable Models. I

A syntactic characterization is furnished for the class of elementary complete theories with finitely many countable models, which is the analog of a known theorem by Ryll-Nardzewski on countably categorical theories, and is based on classifying the theories by Rudin-Keisler quasiorders and distribution functions of a number of models limit over types.     

Countably categorical and autostable Boolean algebras with distinguished ideals

We study countable Boolean algebras with finitely many distinguished ideals (countable I-algebras) whose elementary theory is countably categorical, and autostable I-algebras which form their subclass. We propose a new characterization for the former class that allows to answer a series of questions about the structure of countably categorical and autostable I-algebras.     

Countably categorical weakly o-minimal structures of finite convexity rank

We completely describe countably categorical weakly o-minimal theories of finite convexity rank.     

A countably categorical theory which is not G-compact

We give an example of a countably categorical theory which is not G-compact. The countable model of this theory does not have AZ-enumerations.     

Elementary equivalence of some rings of definable functions

We characterize elementary equivalences and inclusions between von Neumann regular real closed rings in terms of their boolean algebras of idempotents, and prove that their theories are always decidable. We then show that, under some hypotheses, the map sending an L-structure R to the L-structure of definable functions from R ⁿ to R preserves elementary inclusions and equivalences and gives a structure with a decidable theory whenever R is decidable. We briefly consider structures of definable functions satisfying an extra condition such as continuity.      

Preservation of <Emphasis Type="Italic">ω</Emphasis>-Categoricity In Expanding The Models of Weakly <Emphasis Type="Italic">o</Emphasis>-Minimal Theories

Studying the model-theoretic properties that are preserved under expansion of the models of countably categorical weakly o-minimal theories of finite convexity rank with convex unary predicates, we show that countable categoricity and convexity rank are among these properties.     

Stably Definable Classes of Theories

A question is studied as to which properties (classes) of elementary theories can be defined via generalized stability. We present a topological account of such classes. It is stated that some well-known classes of theories, such as strongly minimal, o-minimal, simple, etc., are stably definable, whereas, for instance, countably categorical, almost strongly minimal, ω-stable ones, are not. __________ Translated from Algebra i Logika, Vol. 44, No. 5, pp. 583–600, September–October, 2005. Supported by RFBR grant Nos. 02-01-00540 and 05-01-00411, and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1.     

Some Boolean algebras with finitely many distinguished ideals II

We describe the countably saturated models and prime models (up to isomorphism) of the theory Th_prin of Boolean algebras with a principal ideal, the theory Th_max of Boolean algebras with a maximal ideal, the theory Th_ac of atomic Boolean algebras with an ideal such that the supremum of the ideal exists, and the theory Th_sa of atomless Boolean algebras with an ideal such that the supremum of the ideal exists. We prove that there are infinitely many completions of the theory of Boolean algebras with a distinguished ideal that do not have a countably saturated model. Also, we give a sufficient condition for a model of the theory T_X of Boolean algebras with distinguished ideals to be elementarily equivalent to a countably saturated model of T_X.     

On the Boolean algebras of definable sets in weakly o‐minimal theories

We consider the sets definable in the countable models of a weakly o‐minimal theory T of totally ordered structures. We investigate under which conditions their Boolean algebras are isomorphic (hence T is p‐ω‐categorical), in other words when each of these definable sets admits, if infinite, an infinite coinfinite definable subset. We show that this is true if and only if T has no infinite definable discrete (convex) subset. We examine the same problem among arbitrary theories of mere linear orders. Finally we prove that, within expansions of Boolean lattices, every weakly o‐minimal theory is p‐ω‐categorical. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Decidability of Equational Theories of Coverings of Semigroup Varieties

For every proper semigroup variety X, there exists a semigroup variety Y satisfying the following three conditions: (1) Y covers X, (2) if X is finitely based then so is Y, and (3) the equational theory of X is decidable if and only if so is the equational theory of Y. If X is an arbitrary semigroup variety defined by identities depending on finitely many variables and such that all periodic groups of X are locally finite, then one of the following two conditions holds: (1) all nilsemigroups of X are locally finite and (2) X includes a subvariety Y whose equational theory is undecidable and which has infinitely many covering varieties with undecidable equational theories.     

Abelian-by-G Groups,for G Finite,from the Model Theoretic Point of View

Let G be a finite group. We prove that the theory af abelian-by-G groups is decidable if and only if the theory of modules over the group ring ?[G] is decidable. Then we study some model theoretic questions about abelian-by-G groups, in particular we show that their class is elementary when the order of G is squarefree. Mathematics Subject Classification: 03C60, 03B25.     

A sufficient condition for finite decidability

We prove that a varietyV which is locally finite, finitely generated, congruence permutable and of finite type, and whose subdirectly irreducible algebras are all either abelian or linear type 3 above the monolith is finitely decidable if and only if the theory of the finite abelian algebras inV is decidable.Presented by R. McKenzie.     

An equational theory for a nilpotent <Emphasis Type="Italic">A</Emphasis>-loop

It is shown that a variety generated by a nilpotent A-loop has a decidable equational (quasiequational ) theory. Thereby the question posed by A. I. Mal’tsev in [6] is answered in the negative, and moreover, a finitely presented nilpotent A-loop has a decidable word problem.     

Finitely generated submodels of an uncountably categorical homogeneous structure

We generalize the result of non‐finite axiomatizability of totally categorical first‐order theories from elementary model theory to homogeneous model theory. In particular, we lift the theory of envelopes to homogeneous model theory and develope theory of imaginaries in the case of ω‐stable homogeneous classes of finite U‐rank. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)