http://dspace.nbuv.gov.ua:80/handle/123456789/188574 Tue, 21 Apr 2026 08:36:47 GMT 2026-04-21T08:36:47Z http://dspace.nbuv.gov.ua:80/bitstream/id/564386/ http://dspace.nbuv.gov.ua:80/handle/123456789/188574 http://dspace.nbuv.gov.ua:80/handle/123456789/188682 Structure of relatively free trioids Zhuchok, A.V. Loday and Ronco introduced the notions of a trioid and a trialgebra, and constructed the free trioid of rank 1 and the free trialgebra. This paper is a survey of recent developments in the study of free objects in the varieties of trioids and trialgebras. We present the constructions of the free trialgebra and the free trioid, the free commutative trioid, the free n-nilpotent trioid, the free left (right) n-trinilpotent trioid, and the free rectangular trioid. Some of these results can be applied to constructing relatively free trialgebras. Fri, 01 Jan 2021 00:00:00 GMT http://dspace.nbuv.gov.ua:80/handle/123456789/188682 2021-01-01T00:00:00Z http://dspace.nbuv.gov.ua:80/handle/123456789/188681 Injective stabilization of additive functors, III. Asymptotic stabilization of the tensor product Martsinkovsky, A.; Russell, J. The injective stabilization of the tensor product is subjected to an iterative procedure that utilizes its bifunctor property. The limit of this procedure, called the asymptotic stabilization of the tensor product, provides a homological counterpart of Buchweitz’s asymptotic construction of stable cohomology. The resulting connected sequence of functors is isomorphic to Triulzi’s J-completion of the Tor functor. A comparison map from Vogel homology to the asymptotic stabilization of the tensor product is constructed and shown to be always epic. The category of finitely presented functors is shown to be complete and cocomplete. As a consequence, the inert injective stabilization of the tensor product with fixed variable a finitely generated module over an artin algebra is shown to be finitely presented. Its defect and consequently all right-derived functors are determined. New notions of asymptotic torsion and cotorsion are introduced and are related to each other. Fri, 01 Jan 2021 00:00:00 GMT http://dspace.nbuv.gov.ua:80/handle/123456789/188681 2021-01-01T00:00:00Z http://dspace.nbuv.gov.ua:80/handle/123456789/188680 On the structure of some groups having finite contranormal subgroups Kurdachenko, L.A.; Semko, N.N. Following J.S. Rose, a subgroup H of the group G is said to be contranormal in G, if G = Hᴳ. In a certain sense, contranormal subgroups are antipodes to subnormal subgroups. We study the structure of Abelian-by-nilpotent groups having a finite proper contranormal p-subgroup. Fri, 01 Jan 2021 00:00:00 GMT http://dspace.nbuv.gov.ua:80/handle/123456789/188680 2021-01-01T00:00:00Z http://dspace.nbuv.gov.ua:80/handle/123456789/188679 On extension of classical Baer results to Poisson algebras Kurdachenko, L.A.; Pypka, A.A.; Subbotin I.Ya. In this paper we prove that if P is a Poisson algebra and the n-th hypercenter (center) of P has a finite codimension, then P includes a finite-dimensional ideal K such that P/K is nilpotent (abelian). As a corollary, we show that if the n-th hypercenter of a Poisson algebra P (over some specific field) has a finite codimension and P does not contain zero divisors, then P is an abelian algebra. Fri, 01 Jan 2021 00:00:00 GMT http://dspace.nbuv.gov.ua:80/handle/123456789/188679 2021-01-01T00:00:00Z