Idempotent Elements in a Bernstein Algebra

A finite-dimensional commutative algebra A over a field K is called a Bernstein algebra if there exists a non-trivial homomorphism co: A -> K (baric algebra) such that the identity (x) = CO(X)JC holds in A (see [7]). The origin of Bernstein algebras lies in genetics (see [2,8]). Holgate (in [2]) was the first to translate the problem into the language of non-associative algebras. Information about algebraic properties of Bernstein algebras, as well as their possible genetic interpretations, can be found in [10, Chapter 9B; 11; 12; 3; 1]. The existence of idempotent elements, that is elements e, e # 0, such that e = e, is of interest in the study of the structure of a non-associative algebra. From the biological aspect the existence of such elements is also interesting, because the equilibria of a population which can be described by an algebra correspond to idempotent elements of this algebra. The algebras occurring in applications usually do contain an idempotent. This occurs in Bernstein algebras (see [10]). With respect to an idempotent eeA (whose existence is guaranteed), A splits into the direct sum A = (e) + U+Z, where