  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.
