The purpose of this paper is to introduce and study BiHom-alternative algebras and BiHom-Malcev algebras. It is shown that BiHom-alternative algebras are BiHom-Malcev admissible and BiHom-Jordan admissible. Moreover, BiHom-type generalizations of some well known identities in alternative algebras, including the Moufang identities, are obtained.

In this paper, we define and study (co)homology theories of a compatible associative algebra A. At first, we construct a new graded Lie algebra whose Maurer-Cartan elements are given by compatible associative structures. Then we define the cohomology of a compatible associative algebra A and as applications, we study extensions, deformations and extensibility of finite order deformations of A. We end this paper by considering compatible presimplicial vector spaces and the homology of compatible associative algebras.

In this paper, we introduce the cohomology theory of O-operators on Homassociative algebras. This cohomology can also be viewed as the Hochschild cohomology of a certain Hom-associative algebra with coefficients in a suitable bimodule. Next, we study infinitesimal and formal deformations of an O-operator and show that they are governed by the above-defined cohomology. Furthermore, the notion of Nijenhuis elements associated with an O-operator is introduced to characterize trivial infinitesimal deformations.

No abstract

We introduce the concept of 3-Lie-Rinehart superalgebra and systematically describe a cohomology complex by considering coefficient modules. Furthermore, we study the relationships between a Lie-Rinehart superalgebra and its induced 3-Lie-Rinehart superalgebra. The deformations of 3-Lie-Rinehart superalgebra are considered via the cohomology theory.