# Omsk Algebraic Webinar

Upcoming meeting:

### "Automorphisms of the category of free algebras"

Abstract. There are many various reasons why it is interesting to study automorphisms of some arbitrary algebraic structure. Our considerations are tightly connected with some problems in Universal Algebraic Geometry. The main goal of this talk is give a brief introduction to Universal Algebraic Geometry and to describe the method of verbal operations for a description of automorphisms of the category free finitely generated algebras in a given variety of algebras.

• Preliminary future speakers: Alexander Treyer

Latest announcements:

### "Sublinear time algorithms in (semi)groups"

Abstract. We are going to discuss what can be done in sublinear time (in the "length" of an input); in particular, without reading the whole input but only a small part thereof. One well-known example is deciding divisibility of a decimal integer by $$2$$, $$5$$, or $$10$$: this is done by reading just the last digit. We will discuss some less obvious examples from (semi)group theory.

### "IBN-varieties of algebras"

Abstract. The concept of a variety with IBN (invariant basic number) property first appeared in the ring theory. But we can define this concept for an arbitrary variety of universal algebras with an arbitrary signature. The proof of the IBN propriety of some variety is very important in universal algebraic geometry. This is a milestone in the study of the relation between geometric and automorphic equivalences of algebras of this variety. We prove very simple but very useful for the study of IBN properties of different varieties theorems. We will consider some applications of these theorems. We will consider many-sorted universal algebras as well as one-sorted. So, all concepts and all results will be generalized for the many-sorted case.

### "Equationally noetherian graphs and hypergraphs"

Abstract. A simple graph is called equationally noetherian if any system of equations is equivalent to its finite subsystem. It isn't hard to notice that a simple graph is equationally noetherian if and only if it is equationally noetherian in one variable equations. Based on this property we described all equationally noetherian graphs with help of forbidden subgraphs (joint with Ivan Buchinsky). Also several examples of hypergraphs will be presented that are not equationally noetherian but equationally noetherian in one variable equations.

### "Hardness of equations over finite solvable groups under the exponential time hypothesis"

Abstract. The study of the complexity of the equation satisfiability problem in finite groups had been initiated by Goldmann and Russell in 2002. They showed that this problem is in polynomial time for nilpotent groups, while it is $$NP$$-complete for non-solvable groups. Since then, several results have appeared showing that the problem can be solved in polynomial time in certain solvable groups G having a nilpotent normal subgroup H with nilpotent factor $$G/H$$. In this talk I will show that such normal subgroup must exist in each finite group with equation satisfiability solvable in polynomial time, unless the Exponential Time Hypothesis fails. Moreover, the same hardness result applies to the equation identity problem. The talk is based on joint work with Pawel Idziak, Piotr Kawalek, and Jacek Krzaczkowski.

### "Generic complexity of the identity problem over finite groups and semigroups"

Abstract. The problem of checking identities in algebraic structures is the one of the most fundamental problem in algebra. For finite algebraic structures this problem can be decidable in polynomial time, or hard (co-$$NP$$-complete). In this talk I present generic polynomial algorithms for the identity problem in all finite groups and for some monoids and semigroups. These algorithms work in the cases when the problem is co-$$NP$$-complete.

### "Relative quasivarieties of Hasse diagrams with a single zero"

Abstract. The present work is devoted to the category of Artinian partially ordered sets $$K$$ ($$APOS$$), which is defined as follows: any strictly decreasing chain of elements terminates at a finite number.

### "Centralizer dimension and equationally noetherian groups"

Abstract. In 1999th G.Baumslag, A.Miasnikov and V.Remeslennikov formulated the following problem:

Let $$L$$ be the language of group theory, $$G$$ be a group and $$L_G$$ be the language of group theory with a set of constants from the group $$G$$. Let the group $$G = <G, L_G>$$ be equationally noetherian over one variable equations ($$1$$-equationally noetherian) Does it follow that the group $$G$$ is equationally noetherian?

An example of a nilpotent group will be presented, which is $$1$$-equationally noetherian but is not equationally noetherian in general. The proof will be based on the new results on a close relation between the concepts of centralizer dimension and noetherenes by equations for groups.