The aim of this paper is to investigate the cohomologies for ternary algebras of associative type. We study in particular the cases of partially associative ternary algebras and weak totally associative ternary algebras. Also, we consider the Takhtajan's construction, which was used to construct a cohomology of ternary Nambu-Lie algebras using Chevalley-Eilenberg cohomology of Lie algebras, and discuss it in the case of ternary algebras of associative type. One of the main results of this paper states that a usual deformation cohomology does not exist for partially associative ternary algebras which implies that their operad is not a Koszul operad.

The purpose of the present paper is to consider and study a special form of Rund's h-curvature tensor $K^i_{ljk}$ and Berwald's curvature tensor $H^i_{ljk}$ in an $R3$-like $C$-reducible Finsler space. In this paper, we modify the Rund's h-curvature tensor $K^i_{ljk}$ to special form by using some special Finsler spaces like $C$-reducible, $R3$-like Finsler spaces.

It is known that the Betti numbers of the Heisenberg Lie algebras are unimodal over fields of characteristic two. This note observes that they are log-concave. An example is given of a nilpotent Lie algebra in characteristic two for which the Betti numbers are unimodal but not log-concave.

We construct Lie superalgebras $\mathfrak{osp}(2n+1|4n+2)$ and $\mathfrak{osp}(2n|4n)$ starting with certain classes of anti-structurable algebras via the standard embedding Lie superalgebra construction corresponding to (ε,δ)-Freudenthal Kantor triple systems.

In 1944, R. H. Bruck has described a very general construction method which he called the extension of a set by a quasigroup. We use it to construct a class of examples for LF-quasigroups in which the image of the map $e(x) = x\backslash x$ is a group. More generally, we consider the variety of quasigroups which is defined by the property that the map $e$ is an endomorphism and its subvariety where the image of the map $e$ is a group. We characterize quasigroups belonging to these varieties using their Bruck decomposition with respect to the map $e$.

It is proved that any left F-quasigroup is isomorphic to the direct product of a left F-quasigroup with a unique idempotent element and isotope of a special form of a left distributive quasigroup. The similar theorems are proved for right F-quasigroups, left and right SM- and E-quasigroups. Information on simple quasigroups from these quasigroup classes is given; for example, finite simple F-quasigroup is a simple group or a simple medial quasigroup. It is proved that any left F-quasigroup is isotopic to the direct product of a group and a left S-loop. Some properties of loop isotopes of F-quasigroups (including M-loops) are pointed out. A left special loop is an isotope of a left F-quasigroup if and only if this loop is isotopic to the direct product of a group and a left S-loop (this is an answer to Belousov ``1a'' problem). Any left E-quasigroup is isotopic to the direct product of an abelian group and a left S-loop (this is an answer to Kinyon-Phillips 2.8(1) problem). As corollary it is obtained that any left FESM-quasigroup is isotopic to the direct product of an abelian group and a left S-loop (this is an answer to Kinyon-Phillips 2.8(2) problem). New proofs of some known results on the structure of commutative Moufang loops are presented.