Here we give basic properties of dibaric algebras which are motivated by genetic models. Dibaric algebras are not associative and they have a non trivial homomorphism onto the sex differentiation algebra. We define first join of dibaric algebras next indecomposable dibaric algebras. Finally, we prove the uniqueness of the decomposition of a dibaric algebra, with semiprincipal idempotent, as the join of indecomposable dibaric algebras.

Let A be a commutative algebra over a field F of characteristic = 2, 3. In [M. Gerstenhaber, On nilalgebras and linear varieties of nilpotent matrices II, Duke Math. J. 27 (1960) 21-31], M. Gerstenhaber proved that if A is a nilalgebra of bounded index t and the characteristic of F is zero (or greater than 2t − 3), then the right multiplication R x is nilpotent and R 2t−3 x = 0 for all x ∈ A. In this work, we prove that this result is also valid for commutative power-associative algebras of characteristic t. In Section 3, we prove that when A is a power-associative nilalgebra of dimension 6, then A is nilpotent or (A 2 ) 2 = 0. In Section 4, we prove that every power-associative nilalgebra A of dimension n and nilindex t n − 1 is either nilpotent of index t or isomorphic to the Suttles' example.

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We investigate the structure of commutative non-associative algebras satisfying the identity x x xy = 0. Recently, Correa and Hentzel proved that every commutative algebra satisfying above identity over a field of characteristic = 2 is solvable. We prove that every commutative finite-dimensional algebra over a field F of characteristic = 2 3 which satisfies the identity x x xy = 0 is nilpotent. Furthermore, we obtain new identities and properties for this class of algebras.

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