Problems in algebra inspired by universal algebraic geometry

Abstract

Let Θ be a variety of algebras. In every Θ and every algebra H from Θ one can consider algebraic geometry in Θ over H. We consider also a special categorical invariant K Θ (H) of this geometry. The classical algebraic geometry deals with the variety Θ = Com − P of all associative and commutative algebras over the ground field of constants P . An algebra H in this setting is an extension of the ground field P . Geometry in groups is related to varieties Grp and Grp − G, where G is a group of constants. The case Grp − F where F is a free group, is related to Tarski's problems devoted to logic of a free group.The described general insight on algebraic geometry in different varieties of algebras inspires some new problems in algebra and algebraic geometry. The problems of such kind determine, to a great extent, the content of universal algebraic geometry.For example, a general and natural problem is:When do the algebras H 1 and H 2 have the same geometry?or more specifically,What are the conditions on algebras from a given variety Θ which provide coincidence of their algebraic geometries?We consider two variants of coincidence:2) These categories are equivalent. This problem is highly connected with the following general algebraic problem:Let Θ 0 be the category of all free in Θ algebras W = W (X), where X is finite. Consider the groups of automorphisms Aut (Θ 0 )

Wreath products of semigroups in universal algebraic geometry and Plotkin's problem

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We prove that the wreath product C = A ≀ B of a semigroup A with zero and an infinite cyclic semigroup B is q ω -compact (logically Noetherian). Our result partially solves the Plotkin's problem about wreath products in [6].

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“…The original equation E(X) : t(X) = s(X) defines the equation t B (X) = s B (X) over B, where the terms t B (X), s B (X) are defined by (6,8). By the definition of the set T < , all terms 0,…”

Growth of finitely generated simple Lie algebras

“…We conclude with a remark on a question posed by B. Plotkin. In [20,Problem 6], he asked if there exists a continuum of non-isomorphic finitely generated simple Lie algebras over a fixed base field. It should be mentioned that in [10], Lichtman and Passman answered the analogous question for associative algebras, assuming the base field either contains enough roots of the unity or is countable.…”

Direct products, varieties, and compactness conditions

“…Recall that an algebraic structure A is L-equationally Noetherian if any system of L-equations is equivalent over A to its finite subsystem. The problem about the connections between varieties and equationally Noetherian algebras was posed by B. Plotkin in [7]. Problem 1.…”