We study partially and totally associative ternary algebras of first and second kind. Assuming the vector space underlying a ternary algebra to be a topological space and a~triple product to be continuous mapping, we consider the trivial vector bundle over a ternary algebra and show that a triple product induces a structure of binary algebra in each fiber of this vector bundle. We find the sufficient and necessary condition for a ternary multiplication to induce a structure of associative binary algebra in each fiber of this vector bundle. Given two modules over the algebras with involutions, we construct a~ternary algebra which is used as a building block for a Lie algebra. We construct ternary algebras of cubic matrices and find four different totally associative ternary multiplications of second kind of cubic matrices. It is proved that these are the only totally associative ternary multiplications of second kind in the case of cubic matrices. We describe a ternary analog of Lie algebra of cubic matrices of second order which is based on a notion of j-commutator and find all commutation relations of generators of this algebra.

We study nontrivial deformations of the natural action of the Lie superalgebra $\mathcal{K}(1)$ of contact vector fields on the $(1,1)$-dimensional superspace $\mathbb{R}^{1|1}$ on the space of symbols $\widetilde{{\mathcal{S}}}_\delta^n=\bigoplus_{k=0}^n{\mathfrak{F}}_{\delta-\frac{k}{2}}$. We calculate obstructions for integrability of infinitesimal multiparameter deformations and determine the complete local commutative algebra corresponding to the miniversal deformation.

The notion of color algebras is generalized to the class of $F$-ary algebras, and corresponding decoloration theorems are established. This is used to give a construction of colored structures by means of tensor products with Clifford-like algebras. It is, moreover, shown that color algebras admit realizations as $q=0$ quon algebras.

The cohomology of a~finite-dimensional coalgebra over a finitely generated quadratic operad, with coefficients in itself, is defined and is shown to have the structure of a graded Lie algebra. The cohomology of such a coalgebra is isomorphic to the cohomology of its linear dual as graded Lie algebras.

We introduce the conception of matched pairs of $(H, \beta)$-Lie algebras, and construct an $(H, \beta)$-Lie algebra through them. We prove that the cocycle twist of a~matched pair of $(H, \beta)$-Lie algebras can also be matched.