Cofinal elementary extensions

We investigate some properties of ordered structures that are related to their having cofinal elementary extensions. Special attention is paid to models of some very weak fragments of Peano Arithmetic.     

Cardinal elementary extensions

We determine the consistency strength of some model theoretic extension properties for cardinals.

     

Invariant tori of linear extensions which do not have Green's functions

We study the integral representation of invariant tori of linear extensions which do not have Green's functions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 4, pp. 520–524, April, 1993.     

Typical ambiguity and elementary equivalence

A sentence of the usual language of set theory is said to be stratified if it is obtained by “erasing” type indices in a sentence of the language of Russell's Simple Theory of Types. In this paper we give an alternative presentation of a proof the ambiguity theorem stating that any provable stratified sentence has a stratified proof. To this end, we introduce a new set of ambiguity axioms, inspired by Fraïssé's characterization of elementary equivalence; these axioms can be naturally used to give different proofs of the ambiguity theorem (semantic or syntactic, classical or intuitionistic). MSC: 03B15, 03F50, 03F55.     

Factor orbit equivalence of compact group extensions

LetG be a compact metrizable group. We show that any two ergodic extensions of transformationsT ₁andT ₂ by rotations ofG are factor orbit equivalent relative toT ₁andT ₂, and the equivalence may be taken to have a certain natural form.     

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.     

On elementary equivalence in fuzzy predicate logics

Our work is a contribution to the model theory of fuzzy predicate logics. In this paper we characterize elementary equivalence between models of fuzzy predicate logic using elementary mappings. Refining the method of diagrams we give a solution to an open problem of Hájek and Cintula (J Symb Log 71(3):863–880, 2006, Conjectures 1 and 2). We investigate also the properties of elementary extensions in witnessed and quasi-witnessed theories, generalizing some results of Section 7 of Hájek and Cintula (J Symb Log 71(3):863–880, 2006) and of Section 4 of Cerami and Esteva (Arch Math Log 50(5/6):625–641, 2011) to non-exhaustive models.     

Isomorphism and elementary equivalence of multilinear maps

In [14], we proved that two finitely generated finite-by-nilpotent groups G,H are elementarily equivalent if and only if Z×G and Z×H are isomorphic. In the present paper, we obtain similar characterizations of elementary equivalence for the following classes of structures:

1. the (n+2)-tuples (A ₁…,A _n+1,f),where n≥2 is an integerA ₁…,A _n+1 are disjoint finitely generated abelian groups and f A ₁×…×A _n →A _n+1: is a n-linear map;

2. the triples (A,B f), where n≥2 is an integerA,B are disjoint finitely generated abelian groups and f : A ⁿ →B is a n-linear map;

3. the couples (A,f), where n≥2 is an integerA is a finitely generated abelian group and f:A ⁿ →A is a n-linear map.

For each class, we show that elementary equivalence does not imply isomorphism. In particular, we give an example of two nonisomorphic finitely generated torsion-free Lie rings which are elementarily equivalent.     

Nielsen equivalence classes of free abelianized extensions of groups

Let be a presentation of a group G. We obtain information about the Nielsen equivalence classes of F _n /R′ in terms of the group K ₁(ℤG) that arises in algebraic K-theory. We deduce some results about Nielsen equivalence classes and T-systems of polycyclic groups.     

The Ext-group of unitary equivalence classes of unital extensions

Let A be a unital separable nuclear C*-algebra which belongs to the bootstrap category N and B be a separable stable C*-algebra. In this paper, we consider the group Ext _u (A, B) consisting of the unitary equivalence classes of unital extensions τ: A→Q(B). The relation between Ext _u (A, B) and Ext(A, B) is established. Using this relation, we show the half-exactness of Ext _u (·, B) and the (UCT) for Ext _u (A, B). Furthermore, under certain conditions, we obtain the half-exactness and Bott periodicity of Ext _u (A, ·).     

On elementary equivalence and isomorphism of clone segments

The principal application of a general theorem proved here shows that for any choice 1mnp of integers there exist metric spacesX andY such that the initialk-segments of their clones of continuous maps coincide exactly whenkm, are isomorphic exactly whenkn, and are elementarily equivalent exactly whenkp.Dedicated to Prof. László Fuchs on the occasion of his 70th birthday     

Factor orbit equivalence of compact group extensions and classification of finite extensions of ergodic automorphisms

In §§1–5, we classifyn-point extensions of ergodic automorphisms up to factor orbit-equivalence (which is the natural analogue of factor isomorphism). This classification is in terms of conjugacy classes of subgroups of the symmetric group onn points, and parallels D. Rudolph’s classification ofn-point extensions of Bernoulli shifts up to factor isomorphism. In §6, we give another proof of A. Fieldsteel’s theorem on factor orbit-equivalence of compact group extensions.     

The elementary equivalence of free algebras in cantor varieties

We study properties of free algebras in the Cantor varieties C_m,n. A free algebra of rank r in C_m,n is denoted F_{C m,n}(r). We argue that the following hold: (1) any two C_m,n-free algebras F_{C m,n}(r) and F_{C m,n}(s) of ranks r and s, where r and s are arbitrary (finite or infinite) cardinals, r≥m, and s≥m, are elementary equivalent; (2) any two C_m,n-free algebras F_{C m,n}(r) and F_{C m,n}(s) of ranks r and s, where r and s are arbitrary (finite or infinite) cardinals, are universally equivalent, that is, share one ∀-theory; (3) an elementary theory Th(F_{C m,n}(r)) for an arbitrary C_m,n-free algebra of (finite or infinite) rank r, treated in a signature Ω, is decidable; (4) an elementary theory Th(K) for an arbitrary nonempty class of free algebras in C_m,n, treated in a signature Ω, is decidable. Translated fromAlgebra i Logika, Vol. 38, No. 2, pp. 228–248, March–April, 1999.