Abstract. The aim of this paper is to introduce n-ary Hom-algebra structures generalizing the n-ary algebras of Lie type enclosing n-ary Nambu algebras, n-ary Nambu-Lie algebras, n-ary Lie algebras, and n-ary algebras of associative type enclosing n-ary totally associative and n-ary partially associative algebras. Also, we provide a way to construct examples starting from an n-ary algebra and an n-ary algebras endomorphism. Several examples could be derived using this process.

The aim of this paper is to extend to ternary algebras the classical theory of formal deformations of algebras introduced by Gerstenhaber. The associativity of ternary algebras is available in two forms, totally associative case or partially associative case. To any partially associative algebra corresponds by anti-commutation a ternary Lie algebra. In this work, we summarize the principal definitions and properties as well as classification in dimension 2 of these algebras. Then we focuss ourselves on the partially associative ternary algebras, we construct the first groups of a cohomolgy adapted to formal deformations and then we work out a theory of formal deformation in a way similar to the binary algebras.

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