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We parameterize countable locally standard Boolean measure algebras by pairs of a Steinitz number and a real number greater or equal to 1. This is an analog of the theorems of Dixmier and Baranov. [ABSTRACT FROM AUTHOR]

Let V be an infinite-dimensional vector space over a field of characteristic not equal to 2. We classify ideals of the Lie algebra \mathfrak {gl}(V) of all linear transformations of the space V. [ABSTRACT FROM AUTHOR]

We study an abstract class of Hamming spaces (known also as measure algebras) that generalizes standard Hamming spaces (ℤ / 2 ℤ) n . We classify countable locally standard Hamming spaces and show that each of them can be realized as the Boolean algebra of idempotents of a Cartan subalgebra of a locally matrix algebra. [ABSTRACT FROM AUTHOR]

An F -algebra A with unit 1 is said to be a locally matrix algebra if an arbitrary finite collection of elements a 1 , ... , a s from A lies in a subalgebra B with 1 of the algebra A , that is isomorphic to a matrix algebra M n (F) , n ≥ 1. To an arbitrary unital locally matrix algebra A , we assign a Steinitz number s t (A) and study a relationship between s t (A) and A. [ABSTRACT FROM AUTHOR]

For an arbitrary finite metric space (X, d) a subset A, A ⊂ X, is called a resolving set if for any two points x and y from the space X there is an element a from subset A, such that distances d(a, x) and d(a, y) are different. The metric dimension md(X) of the space X is the minimum cardinality of a resolving set. It is well known that the problem of finding the metric dimension of a metric space is NP-complete [7]. In this paper, the metric dimension for finite ultrametric spaces is completely characterized. It is proved that for any finite ultrametric space there exists a polynomial-time algorithm for determining the metric dimension of this spaces. [ABSTRACT FROM AUTHOR]

We construct a unital locally matrix algebra of uncountable dimension that (1) does not admit a primary decomposition, (2) has an infinite locally finite Steinitz number. It gives negative answers to questions from [V. M. Kurochkin, On the theory of locally simple and locally normal algebras, Mat. Sb., Nov. Ser. 22(64)(3) (1948) 443-454; O. Bezushchak and B. Oliynyk, Unital locally matrix algebras and Steinitz numbers, J. Algebra Appl. (2020), online ready]. We also show that for an arbitrary infinite Steinitz number s there exists a unital locally matrix algebra A having the Steinitz number s and not isomorphic to a tensor product of finite-dimensional matrix algebras. [ABSTRACT FROM AUTHOR]

The construction of homogeneous monomial groups are given and their basic properties are studied. The structure of a centralizer of an element is completely described and the problem of conjugacy of two elements is resolved. Moreover, the classification of homogeneous monomial groups are determined by using the lattice of Steinitz numbers, namely, we prove the following: Let λ andμbe two Steinitz numbers. The homogeneous monomial groupsΣλ(H) andΣμ(G) are isomorphic if and only if λ = μandH≅Gprovided that the splittings ofΣλ(H) andΣμ(G) are regular. [ABSTRACT FROM AUTHOR]

We consider isometry groups of a fairly general class of non standard products of metric spaces. We present sufficient conditions under which the isometry group of a non standard product of metric spaces splits as a permutation group into direct or wreath product of isometry groups of some metric spaces. [ABSTRACT FROM AUTHOR]

In this paper semigroups of contractions of metric spaces are considered. The semigroup of contractions of the wreath product of metric spaces is calculated. [ABSTRACT FROM AUTHOR]