ON SEPARABILITY FINITENESS CONDITIONS IN SEMIGROUPS

Taking residual finiteness as a starting point, we consider three related finiteness properties: weak subsemigroup separability, strong subsemigroup separability and complete separability. We investigate whether each of these properties is inherited by Schützenberger groups. The main result of this paper states that for a finitely generated commutative semigroup S, these three separability conditions coincide and are equivalent to every $\mathcal {H}$ -class of S being finite. We also provide examples to show that these properties in general differ for commutative semigroups and finitely generated semigroups. For a semigroup with finitely many $\mathcal {H}$ -classes, we investigate whether it has one of these properties if and only if all its Schützenberger groups have the property.

Galois coverings of one-sided bimodule problems

Applying geometric methods of 2-dimensional cell complex theory, we construct a Galois covering of a bimodule problem satisfying some structure, triangularity and finiteness conditions in order to describe the objects of finite representation type. Each admitted bimodule problem A is endowed with a quasi multiplicative basis. The main result shows that for a problem from the considered class having some finiteness restrictions and the schurian universal covering A', either A is schurian, or its basic bigraph contains a dotted loop, or it has a standard minimal non-schurian bimodule subproblem.

On separability properties in direct products of semigroups

We investigate four finiteness conditions related to residual finiteness: complete separability, strong subsemigroup separability, weak subsemigroup separability and monogenic subsemigroup separability. For each of these properties we examine under which conditions the property is preserved under direct products. We also consider if any of the properties are inherited by the factors in a direct product. We give necessary and sufficient conditions for finite semigroups to preserve the properties of strong subsemigroup separability and monogenic subsemigroup separability in a direct product.

Hochschild cohomology, finiteness conditions and a generalization of d-Koszul algebras

Given a finite-dimensional algebra [Formula: see text] and [Formula: see text], we construct a new algebra [Formula: see text], called the stretched algebra, and relate the homological properties of [Formula: see text] and [Formula: see text]. We investigate Hochschild cohomology and the finiteness condition (Fg), and use stratifying ideals to show that [Formula: see text] has (Fg) if and only if [Formula: see text] has (Fg). We also consider projective resolutions and apply our results in the case where [Formula: see text] is a [Formula: see text]-Koszul algebra for some [Formula: see text].

On Periodic Shunkov’s Groups with Almost Layer-finite Normalizers of Finite Subgroups

Layer-finite groups first appeared in the work by S. N. Chernikov (1945). Almost layer-finite groups are extensions of layer-finite groups by finite groups. The class of almost layer-finite groups is wider than the class of layer-finite groups; it includes all Chernikov groups, while it is easy to give examples of Chernikov groups that are not layer-finite. The author develops the direction of characterizing well-known and well-studied classes of groups in other classes of groups with some additional (rather weak) finiteness conditions. A Shunkov group is a group 𝐺 in which for any of its finite subgroups 𝐾 in the quotient group <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msub><mi>N</mi><mi>G</mi></msub><mo>(</mo><mi>K</mi><mo>)</mo></mrow><mi>K</mi></mfrac></math> any two conjugate elements of prime order generate a finite subgroup. In this paper, we prove the properties of periodic not almost layer-finite Shunkov groups with condition: the normalizer of any finite nontrivial subgroup is almost layer-finite. Earlier, these properties were proved in various articles of the author, as necessary, sometimes under some conditions, then under others (the minimality conditions for not almost layer-finite subgroups, the absence of second-order elements in the group, the presence of subgroups with certain properties in the group). At the same time, it was necessary to make remarks that this property is proved in almost the same way as in the previous work, but under different conditions. This eliminates the shortcomings in the proofs of many articles by the author, in which these properties are used without proof.

Stable systems of schedule contracts

The paper studies the systems of paired contracts between agents of two complementary groups (workers and firms, students and universities, depositors and banks). Multiple contracts are allowed, as well as flexible contracts when the contract is concluded with some intensity. Agent preferences are described using choice functions. It is shown that if these choice functions satisfy the condition that we call conservativeness, then there are so-called stable systems of contracts, when it is unprofitable for any pair of counterparties to change the concluded contracts. The existence of a stable system of contracts is established using the transfinite process of sequential approximation, which generalizes the classical Gale−Shapley algorithm. As a result, we relieved of the finiteness conditions of the set of contracts. Such properties of stable systems as polarization and latticeness are also studied.