# Omsk Algebraic Online Webinar

### "Which properties of groups can be defined by their pregroup structures?"

Abstract. In this talk we discuss which properties of pregroups translate into the corresponding properties of their universal groups. First, we are going to give a survey of known results on the subject and then report some new results from the ongoing research on pregroup structures encoding hyperbolicity.

### "Rich algebraic structures and weak second order logic"

Abstract. "What can be described by first-order formulas in a given algebraic structure $$A$$?" - an old and interest question. Of course, much depends on the structure $$A$$. For example, in a free group, only cyclic subgroups (and the group itself) are definable in the first order logic, but in a free monoid of finite rank we can define any finitely generated submonoid. An algebraic structure $$A$$ is called rich if the first order logic in $$A$$ is equivalent to weak second order logic.

Surprising but there are many interesting groups, rings, and semigroups that are rich. I will talk about some of them, and then I will describe various algebraic, geometric and algorithmic properties that we able to describe by formulas in these systems.

Weak second order logic can be presented in algebraic systems in different ways: through HF-logic, or list superstructures over $$A$$, or recursively enumerable (arithmetic) infinite disjunctions and conjunctions, or via finite binary predicates, etc. I will describe some particular form of this logic that is especially convenient for use in algebra, and show how to effectively translate such weak second-order formulas into equivalent first-order formulas in the case of a rich structure $$A$$.

### "Direct powers of algebraic structures in algebraic geometry"

Abstract. In this talk I begin with elementary properties of equations over direct powers of algebraic structures. We discuss the Noetherian property for direct powers of groups, semigroups, graphs etc. In the conclusion I present a general result asserting that a direct power PA of any finite algebraic structure A is weakly equationally Noetherian (i.e. any system of equations with constants over PA is equivalent to some finite system).

### "Automorphic equivalence in the classical varieties of linear algebras"

Abstract. We consider some variety of universal algebras $$\Theta$$ and the category $$\Theta^0$$. Objects of this category are finitely generated free algebras of the variety $$\Theta$$, morphisms of this category are homomorphisms of these algebras.
After this we consider algebras $$H_1, H_2 \in \Theta$$. The automorphisms of the category $$\Theta^0$$ are very important in the study of the question when these algebras have same algebraic geometry.
In this talk we will consider as $$\Theta$$ the classical varieties of linear algebras:

1. Variety of all linear algebras,
2. Variety of all commutative algebras,
3. Variety of all power associative algebras,
4. Variety of all alternative algebras,
5. Variety of all associative algebras,
6. Variety of all Jordan algebras,
7. Arbitrary subvariety of the variety of all anticommutative algebras over the arbitrary field k of characteristic $$0$$.

The structure of the group of the all automorphisms of the category $$\Theta^0$$ will be studied in all these cases. Also examples of algebras which are not geometric equivalent (the families of closed congruences are not coincides) but have the same algebraic geometry (exist a monotone bijection between these families) will be given in all these cases.

### "Structures with non-standard Kleene stars"

Abstract. Kleene iteration, or Kleene star, is one of the most intriguing algebraic operations appearing in theoretical computer science. Usually, the Kleene star is interpreted as the limit of $$n$$-th powers of an element. This is how it behaves in algebras of formal languages and in relational algebras. In the talk, we shall show some examples of structures with non-standard Kleene iteration, and apply them to proving some results for corresponding logical theories.

### "Birkhoff conjecture and angular billiards"

Abstract. We discuss Birkhoff conjecture on integrable billiards inside convex smooth curves on the plane.

### "Central elements of the distribution algebra of a general linear supergroup and supersymmetric elements"

Abstract. We discuss the notion of supersymmetric element, recently introduced in the joint work with F.Marko. We also show how it relates to the description of the center of the distribution algebra of a general linear supergroup as well as to the description of blocks of the category of supermodules over a general linear supergroup.

### "Automorphisms of the category of finitely generated free groups of the certain subvariety of the variety of all groups"

Abstract. In universal algebraic geometry the category $$Θ^0$$ of finitely generated free algebras of some fixed variety Θ of algebras and the quotient group $$A/I$$ are very important. Here $$A$$ is the group of automorphisms of the category $$Θ^0$$ and $$I$$ is the group of inner automorphisms of this category.

In the varieties of all groups, of all abelian groups (B. Plotkin, G. Zhitomirski, "On automorphisms of categories of free algebras of some varieties", Journal of Algebra. 306:2 (2006), DOI: 10.1016/j.jalgebra.2006.07.028), of all nilpotent groups of the class no more than $$n$$ ($$n ≥ 2$$) (A. Tsurkov, "Automorphisms of the category of the free nilpotent groups of the fixed class of nilpotency", International Journal of Algebra and Computation, 17(5/6) (2007), DOI: 10.1142/S021819670700413X.1273—1281) the group $$A/I$$ is trivial. B. Plotkin posed the following question: "Is there a subvariety of the variety of all groups, such that the group $$A/I$$ in this subvariety is not trivial?" A. Tsurkov hypothesized that for some varieties of periodic groups the group $$A/I$$ is not trivial. In this talk we give an example of one particular subvariety of this kind.

### "Generic complexity of two problems about semigroups"

Abstract. First problem is the word problem for some finitely defined semigroups. In 2008 Won suggested a simple generic algorithm for the word problem in finitely defined semigroups. It works for classical semigroups with undecidable word problem: Tseitin semigroup, Makanin semigroup. But it does not work for semigroups with one relation. The problem of existence of algorithms for word problem in these semigroups is still open, despite the efforts of Adjan and his students. In this talk I present a polynomial generic algorithm for some finitely defined semigroups, which works for Tseitin semigroups and semigroups with one relation.

Second problem is the isomorphism problem for finite semigroups, represented by multiplication tables. Kornienko, Zinchenko and Tyshkevich in 1982 proved that the famous graph isomorphism problem can be reduced in polynomial time to the isomorphism problem for finite semigroups. Thus it is still unknown, does it decidable in polynomial time. In this talk I will present a polynomial generic algorithm for the isomorphism problem for finite semigroups.

### "Residually finite and equationally Noetherian groups"

Abstract. We say a group G is residually finite ($$RF$$) if any non-trivial element of $$G$$ is non-trivial in some finite quotient, and equationally Noetherian ($$EN$$) if every system of equations in $$G$$ is equivalent to a finite subsystem. These two properties, although a priori unrelated, turn out to exhibit similar behaviour, especially among finitely generated groups. For instance, all finitely generated linear groups and rigid soluble groups are both $$RF$$ and $$EN$$, certain wreath products and non-Hopfian groups are neither, and both of these properties are stable under taking subgroups, direct and free products, and finite extensions. Until recently, we did not seem to know any explicit examples of finitely generated groups that are $$RF$$ but not $$EN$$, or vice versa.

In this talk, I will compare and contrast the classes of finitely generated $$RF$$ and $$EN$$ groups. On the one hand, I will illustrate the similarities between these two classes by discussing their shared properties and introducing several "natural" common subclasses. On the other hand, I will give easy examples of finitely generated groups that are $$RF$$ but not $$EN$$, and vice versa.

### "Description of the coordinate groups of irreducible algebraic sets over free $$2$$-nilpotent groups"

Abstract. The talk is based on a joint work with M. Amaglobeli (Tbilisi State university) and V. Remeslennikov (Sobolev Institute of Mathematics).

We give a pure algebraic description of the coordinate groups of irreducible algebraic sets over non-abelian free $$2$$-nilpotent group $$N$$. As a corollary we describe finitely generated groups $$H$$ which are universally equivalent to the group $$N$$ (with constants from $$N$$ in the language). Besides, we give a pure algebraic criterion when a group $$H$$, containing $$N$$ as a subgroup, and $$N$$-separated by $$N$$, is in fact $$N$$-discriminated by $$N$$.

### "Geometrical Equivalence and Action Type Geometrical Equivalence of Group Representations"

Abstract. The universal algebraic geometry of group representations was considered in: B. Plotkin, A. Tsurkov, “Action type geometrical equivalence of group representations”, Algebra and discrete mathematics, 4: (2005), 48-79. The concepts of geometrical equivalence and action type geometrical equivalence of group representations were defined. It was proved that if two representations are geometrically equivalent then they are action type geometrically equivalent. Also it was remarked that if two representations $$(V_1,G_1)$$ and $$(V_2,G_2)$$ are action type geometrically equivalent and groups G₁ and G₂ are geometrically equivalent, the representations $$(V_1,G_1)$$ and $$(V_2,G_2)$$ are not necessarily geometrically equivalent. However some specific counterexample was not presented. In this talk we fill the gap and construct the example of two representations $$(V_1,G_1)$$ and $$(V_2,G_2)$$ which are action type geometrically equivalent and groups $$G_1$$ and $$G_2$$ are geometrically equivalent, but the representations $$(V_1,G_1)$$ and $$(V_2,G_2)$$ are not geometrically equivalent.

### "Plotkin`s problem for semigroups"

Abstract. B.Plotkin posed the following problem: when a wreath product of two groups is $$q_{\omega}$$-compact or equationally Noetherian? We solve this problem for wreath products of semigroups $$A$$ wr $$B$$ where $$B$$ is infinite cyclic and $$A$$ contains zero.

### "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.