On some model theoretic problems concerning certain extensions of abelian groups by groups of finite exponent
On the Rank of Ext.
On universal theories of metabelian groups and the Shmel'kin embedding.
ℳ-rank and meager groups
Assume p* is a meager type in a superstable theory T. We investigate definability properties of p*-closure. We prove that if T has countable models then the multiplicity rank ℳ of every type p is finite. We improve Saffe’s conjecture.
Realizations of Loops and Groups defined by short identities
In a recent paper, those quasigroup identities involving at most three variables and of “length” six which force the quasigroup to be a loop or group have been enumerated by computer. We separate these identities into subsets according to what classes of loops they define and also provide humanly-comprehensible proofs for most of the computer-generated results.
Relative geometries
In this paper, we shall study type-definable groups in a simple theory with respect to one or several stable reducts. While the original motivation came from the analysis of definable groups in structures obtained by Hrushovski's amalgamation method, the notions introduced are in fact more general, and in particular can be applied to certain expansions of algebraically closed fields by operators.
Remarks to Głazek's results on n-ary groups
This is a survey of the results obtained by K. Głazek and his co-workers. We restrict our attention to the problems of axiomatizations of n-ary groups, classes of n-ary groups, properties of skew elements and homomorphisms induced by skew elements, constructions of covering groups, classifications and representations of n-ary groups. Some new results are added too.
Right division in Moufang loops
If is a group, and the operation is defined by then by direct verification is a quasigroup which satisfies the identity . Conversely, if one starts with a quasigroup satisfying the latter identity the group can be constructed, so that in effect is determined by its right division operation. Here the analogous situation is examined for a Moufang loop. Subtleties arise which are not present in the group case since there is a choice of defining identities and the identities produced by...
Rigidity, internality and analysability
We prove a version of Hrushovski's Socle Lemma for rigid groups in an arbitrary simple theory.
Small profinite m-stable groups
A small profinite m-stable group has an open abelian subgroup of finite ℳ-rank and finite exponent.
Some model theory of simple algebraic groups over algebraically closed fields
Some structure theorems for inverse ω-semigroups
Strong boundedness and algebraically closed groups
Let be a non-trivial algebraically closed group and be a subset of generating in infinitely many steps. We give a construction of a binary tree associated with . Using this we show that if is -existentially closed then it is strongly bounded.
Subgroup criteria for an abelian group.
Tarski's problem about the elementary theory of free groups has a positive solution.
The combinatorial derivation and its inverse mapping
Let G be a group and P G be the Boolean algebra of all subsets of G. A mapping Δ: P G → P G defined by Δ(A) = {g ∈ G: gA ∩ A is infinite} is called the combinatorial derivation. The mapping Δ can be considered as an analogue of the topological derivation d: P X→ P X, A ↦ A d, where X is a topological space and A d is the set of all limit points of A. We study the behaviour of subsets of G under action of Δ and its inverse mapping ∇. For example, we show that if G is infinite and I is an ideal in...
The elementary theory of Abelian groups with m-chains of pure subgroups
The homomorphic images of infinite symmetric groups.
Theorems on groups of finite Morley rank analogous to theorems on finite groups.
Torsion-free constructive nilpotent -groups.