ABSTRACT
Baric algebras play a central role in the theory of genetic algebras. They were introduced by I.M.H. Etherington [6], aiming for an algebraic treatment of population genetics. But the whole class of baric algebras is too large, some conditions (usually with a background in genetics) must be imposed in order to obtain a workable mathematical object. With this in mind, several classes of baric algebras have been defined: train, Bernstein, special triangular, etc. As a sample of the work in the field of genetic algebra, see [14] and [15]. In this section we will give the basic definitions on the theory of the baric
algebras, some of them appear in [1], [2], [9], [14] and [15]. Let F be a field, A an algebra over F , not necessarily associative, com-
mutative or finite dimensional and ω : A→ F be a nonzero homomorphism. The ordered pair (A,ω) will be called a baric algebra over F and ω its weight function. If B ⊆ A, we denote by bar(B) the set of all x ∈ B such that ω(x) = 0,
that is, B ∩ kerω. If I ⊆ bar(A) is a two-sided ideal of A, then I is called a b-ideal of A. If e ∈ A is such that e2 = e and ω(e) = 1 then e is called idempotent of weight 1. A b-homomorphism from (A,ω) to (A′, ω′) is a homomorphism of F -algebras ϕ : A → A′ such that ω′ ◦ ϕ = ω. We will denote (A,ω) ∼=b (A′, ω′) or A ∼=b A′, when there exists a b-isomorphism from (A,ω) to (A′, ω′).
