New examples of small Polish structures

We answer some questions from a paper of Krupi\'nski by giving suitable examples of small Polish structures. First, we present a class of small Polish group structures without generic elements. Next, we construct a first example of a small non-zero-dimensional Polish $G$-group

On \omega-categorical, generically stable groups

We prove that each \omega-categorical, generically stable group is solvable-by-finite.Comment: 11 page

Left-ordered inp-minimal groups

We prove that any left-ordered inp-minimal group is abelian, and we provide an example of a non-abelian left-ordered group of dp-rank 2

Locally finite profinite rings

We investigate the structure of locally finite profinite rings. We classify (Jacobson-) semisimple locally finite profinite rings as products of complete matrix rings of bounded cardinality over finite fields, and we prove that the Jacobson radical of any locally finite profinite ring is nil of finite nilexponent. Our results apply to the context of small compact $G$-rings, where we also obtain a description of possible actions of $G$ on the underlying ring.Comment: 17 page

The Lascar groups and the 1st homology groups in model theory

Let $p$ be a strong type of an algebraically closed tuple over B=\acl^{\eq}(B) in any theory $T$. Depending on a ternary relation \indo^* satisfying some basic axioms (there is at least one such, namely the trivial independence in $T$), the first homology group $H^*_1(p)$ can be introduced, similarly to \cite{GKK1}. We show that there is a canonical surjective homomorphism from the Lascar group over $B$ to $H^*_1(p)$. We also notice that the map factors naturally via a surjection from the `relativised' Lascar group of the type (which we define in analogy with the Lascar group of the theory) onto the homology group, and we give an explicit description of its kernel. Due to this characterization, it follows that the first homology group of $p$ is independent from the choice of \indo^*, and can be written simply as $H_1(p)$. As consequences, in any $T$, we show that $|H_1(p)|\geq 2^{\aleph_0}$ unless $H_1(p)$ is trivial, and we give a criterion for the equality of stp and Lstp of algebraically closed tuples using the notions of the first homology group and a relativised Lascar group. We also argue how any abelian connected compact group can appear as the first homology group of the type of a model.Comment: 30 pages, no figures, this merged with the article arXiv:1504.0772

Topologies induced by group actions

We introduce some canonical topologies induced by actions of topological groups on groups and rings. For $H$ being a group [or a ring] and $G$ a topological group acting on $H$ as automorphisms, we describe the finest group [ring] topology on $H$ under which the action of $G$ on $H$ is continuous. We also study the introduced topologies in the context of Polish structures. In particular, we prove that there may be no Hausdorff topology on a group $H$ under which a given action of a Polish group on $H$ is continuous.Comment: 13 page

Sets, groups, and fields definable in vector spaces with a bilinear form

We study definable sets, groups, and fields in the theory $T_\infty$ of infinite-dimensional vector spaces over an algebraically closed field equipped with a nondegenerate symmetric (or alternating) bilinear form. First, we define an ($\mathbb{N}\times \mathbb{Z},\leq_{lex}$)-valued dimension on definable sets in $T_\infty$ enjoying many properties of Morley rank in strongly minimal theories. Then, using this dimension notion as the main tool, we prove that all groups definable in $T_\infty$ are (algebraic-by-abelian)-by-algebraic, which, in particular, answers a question of Granger. We conclude that every infinite field definable in $T_\infty$ is definably isomorphic to the field of scalars of the vector space. We derive some other consequences of good behaviour of the dimension in $T_\infty$, e.g. every generic type in any definable set is a definable type; every set is an extension base; every definable group has a definable connected component. We also consider the theory $T^{RCF}_\infty$ of vector spaces over a real closed field equipped with a nondegenerate alternating bilinear form or a nondegenerate symmetric positive-definite bilinear form. Using the same construction as in the case of $T_\infty$, we define a dimension on sets definable in $T^{RCF}_\infty$, and using it we prove analogous results about definable groups and fields: every group definable in $T^{RCF}_{\infty}$ is (semialgebraic-by-abelian)-by-semialgebraic (in particular, it is (Lie-by-abelian)-by-Lie), and every field definable in $T^{RCF}_{\infty}$ is definable in the field of scalars, hence it is either real closed or algebraically closed.Comment: v2: The particular bounds on dimension obtained in Section 3 were corrected, and a number of minor corrections has been made throughout the pape

The Homology Relation between Molecules: a Revival of an Old Way for Classification of Molecules

The homology (homolo) relation between molecules was introduced. This relation is a generalization of an old idea of series of homologous compounds. The homolo relation operates on a molecule by removing all the structural fragments that are identical with a certain selected fragment. As a result a multiset of fragments is produced. It was shown that the homolo relation is an equivalence relation in a set of molecules. Thus, by choosing various reference fragments, the molecules can be classified in many different ways. Using the language of homolo operation it is possible to redefine such ideas as constitutional and stereo isomers as well as a generator of a molecule and, for instance, factorization of a molecule onto fragments. </p