Elementary Equivalence of Linear Groups Over Rings with a Finite Number of Central Idempotents and Over Boolean Rings

In the present paper, we start with a criterion of elementary equivalence of linear groups over rings with just a finite number of central idempotents. Then we study elementary equivalence of linear groups over Boolean algebras. We prove that two linear groups over Boolean algebras are elementarily equivalent if and only if their dimensions coincide and these Boolean algebras are elementarily equivalent.     

Groups elementarily equivalent to a free 2-nilpotent group of finite rank

We give a characterization of groups elementarily equivalent to a free 2-nilpotent group of finite rank. Translated from Algebra i Logika, Vol. 48, No. 2, pp. 203–244, March–April, 2009.     

Two remarks on the first-order theories of Baumslag-Solitar groups

We characterize all finitely generated groups elementarily equivalent to a solvable Baumslag-Solitar group BS(m, 1). It turns out that a finitely generated group G is elementarily equivalent to BS(m, 1) if and only if G is isomorphic to BS(m, 1). Furthermore, we show that two Baumslag-Solitar groups are existentially (universally) equivalent if and only if they are elementarily equivalent if and only if they are isomorphic.     

Elementary equivalence of incidence rings over semi-perfect rings

In this paper, we prove that if two incidence rings constructed by the same semiperfect ring and some two quasi-ordered sets are elementarily equivalent, then the given sets are elementarily equivalent.     

Elementary equivalence of semigroups of invertible matrices with nonnegative elements over commutative partially ordered rings

In the paper, we prove that if two semigroups of invertible matrices with nonnegative elements over partially ordered commutative rings are elementarily equivalent, then their dimensions coincide and the corresponding semirings of nonnegative elements are elementarily equivalent.     

Automorphisms of the semigroup of invertible matrices with nonnegative elements over commutative partially ordered rings

In the paper, we prove that if two semigroups of invertible matrices with nonnegative elements over partially ordered commutative rings are elementarily equivalent, then their dimensions coincide and the corresponding semirings of nonnegative elements are elementarily equivalent.     

The low<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> and low<Subscript><Emphasis Type="Italic">m</Emphasis></Subscript> r. e. degrees are not elementarily equivalent

Jockusch, Li and Yang showed that the Lown and Low1 r.e. degrees are not elementarily equivalent for n > 1. We answer a question they raise by using the results of Nies, Shore and Slaman to show that the Lown and Lowm r.e. degrees are not elementarily equivalent for n > m > 1.     

Elementary equivalence of the semigroup of invertible matrices with nonnegative elements

In this paper, we prove that the semigroups of invertible matrices with nonnegative elements over linearly ordered associative rings are elementarily equivalent if and only if the matrices have the same dimension and the rings are elementarily equivalent as ordered rings. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 2, pp. 39–53, 2006.     

Isotypeness of Models and Knowledge Bases Equivalence

The aim of this work is to study the notions of elementarily equivalent and isotypic knowledge bases. We prove that isotypic knowledge bases are informationally equivalent.     

Self-injective regular algebras and function rings

An example is given of a self-injective commutative von Neumann regular algebra over the rationals, that is not elementarily equivalent to a function ring.Presented by S. Burris.     

Cancellation and elementary equivalence of groups

If A and B are groups such that $\text{A x}\text{Z}\text{? B x}\text{Z}$, then A and B are elementarily equivalent. From this follows the existence of finitely generated torsion-free nilpotent groups which are elementarily equivalent without being isomorphic.     

Elementary equivalence of unitary linear groups over rings

We find necessary and sufficient conditions for two unitary linear groups over rings with involution with forms of hyperbolic rank at least 2 to be elementarily equivalent.     

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.     

Typical equivalence of linear groups and other algebraic systems

The paper is devoted to the notion of typical equivalence introduced by B. I. Plotkin. We give some examples of elementarily equivalent objects that are not typically equivalent and show two ways to construct nonisomorphic typically equivalent algebras. We also prove A. I. Maltsev??s theorem on elementary equivalence of linear groups over fields for the case of typical equivalence.     

The commutator width of some relatively free lie algebras and nilpotent groups

We determine the exact values of the commutator width of absolutely free and free solvable Lie rings of finite rank, as well as free and free solvable Lie algebras of finite rank over an arbitrary field. We calculate the values of the commutator width of free nilpotent and free metabelian nilpotent Lie algebras of rank 2 or of nilpotency class 2 over an arbitrary field. We also find the values of the commutator width for free nilpotent and free metabelian nilpotent Lie algebras of finite rank at least 3 over an arbitrary field in the case that the nilpotency class exceeds the rank at least by 2. In the case of free nilpotent and free metabelian nilpotent Lie rings of arbitrary finite rank, as well as free nilpotent and free metabelian nilpotent Lie algebras of arbitrary finite rank over the field of rationals, we calculate the values of commutator width without any restrictions. It follows in particular that the free or nonabelian free solvable Lie rings of distinct finite ranks, as well as the free or nonabelian free solvable Lie algebras of distinct finite ranks over an arbitrary field are not elementarily equivalent to each other. We also calculate the exact values of the commutator width of free ?-power nilpotent, free nilpotent, free metabelian, and free metabelian nilpotent groups of finite rank.     

Structure of the Algebraic Systems with the Complete Theory of Infinite Subsystems

The structure is described for the algebraic systems with the complete theory of infinite subsystems, i.e., the systems with all infinite subsystems elementarily equivalent. A theorem is proved characterizing the property of a system to have exactly the sole complete theory of infinite subsystems.     

Every two elementarily equivalent models have isomorphic ultrapowers

We prove (without G.C.H.) that every two elementarily equivalent models have isomorphic ultrapowers, and some related results. The preparation of this paper was supported in Part by NSF Grant GP-22937.     

Henselianity in the language of rings

We consider four properties of a field K related to the existence of (definable) henselian valuations on K and on elementarily equivalent fields and study the implications between them. Surprisingly, the full pictures look very different in equicharacteristic and mixed characteristic.     

Term algebras of elementarily equivalent atom structures

We exhibit two relation algebra atom structures such that they are elementarily equivalent but their term algebras are not. This answers Problem 14.19 in Hirsch and Hodkinson’s text Relation Algebras by Games.