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The main objective of this paper is to show that the notion of type which was developed within the frames of logic and model theory has deep ties with geometric properties of algebras. These ties go back and forth from universal algebraic geometry to the model theory through the machinery of algebraic logic. We show that types appear naturally as logical… (More)
We characterise the class of finite solvable groups by two-variable identities in a way similar to the characterisation of finite nilpotent groups by Engel identities. Let u1 = x−2y−1x, and un+1 = [xunx−1, yuny−1]. The main result states that a finite group G is solvable if and only if for some n the identity un(x, y) ≡ 1 holds in G. We also develop a new… (More)
We characterise the solvable groups in the class of finite groups by an inductively defined sequence of two-variable identities. Our main theorem is the analogue of a classical theorem of Zorn which gives a characterisation of the nilpotent groups in the class of finite groups by a sequence of two-variable identities. To cite this article: T. Bandman et… (More)
We are looking for the smallest integer k > 1 providing the following characterization of the solvable radical R.G/ of any finite group G: R.G/ coincides with the collection of all g 2 G such that for any k elements a1; a2; : : : ; ak 2 G the subgroup generated by the elements g; aiga 1 i , i D 1; : : : ; k, is solvable. We consider a similar problem of… (More)
In the paper we pose and discuss new Burnside-type problems, where the role of nilpo-tency is replaced by that of solvability.
We prove that the solvable radical of a finite group G coincides with the set of elements y having the following property: for any x ∈G the subgroup of G generated by x and y is solvable. This confirms a conjecture of Flavell. We present analogues of this result for finite-dimensional Lie algebras and some classes of infinite groups. We also consider a… (More)
We prove that for a given element P (X1, . . . , Xd) of the finitely generated free Lie algebra Ld, the induced map P : g → g is dominant for any Chevalley algebra g, provided that K is of characteristic 6= 2, and P is not an identity in sl(2,K). We prove that for the Engel monomials [[[X,Y ], Y ], . . . , Y ] and for their linear combinations this map is… (More)
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. Mathematics Subject Classification (2010). Primary 08A70; Secondary 68T30.
We are looking for the smallest integer k > 1 providing the following characterization of the solvable radical R(G) of any finite group G: R(G) consists of the elements g such that for any k elements a1, a2, . . . , ak ∈G the subgroup generated by the elements g,aiga −1 i , i = 1, . . . , k, is solvable. Our method is based on considering a similar problem… (More)