B-algebras and generalized alternative algebras II Baric algebras play a central role in the theory of genetic algebras. They were introduced by I. M. H. Etherington, in [2], in order to give an algebraic treatment to Genetic Populations. Several classes of b-algebras have been defined, such as: train, Bernstein, special triangular, etc. Let U be an algebra over a field F not necessarily associative, commutative or finite dimensional. If ω : U −→ F is a nonzero homomorphism of algebras, then

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... ordered pair (U, ω) will be called a b-algebra over F and ω its weight function or simply its weight. For When B is a subalgebra of U and B 6 ⊂ kerω, then B is called a bsubalgebra of (U, ω). In this case, . If B is a b-subalgebra of U and bar(B) is a two-side ideal of bar(U ) (then by [2, Proposition 1.1]), it is also a two-sided ideal of U ), then B is called normal b-subalgebra of (U, ω). A nonzero element e ∈ U is called an idempotent if e 2 = e and nontrivial idempotent if it is an idempotent different from multiplicative identity element, if the algebra has this element. If (U, ω) is a b-algebra and e ∈ U is an idempotent, then ω(e) = 0 or ω(e) = 1. When ω(e) = 1, then e is called idempotent of weight 1. A b-algebra (U, ω) is called b-simple if for all normal b-subalgebra B of U, bar(B) = (0) or bar(B) = bar(U ). When (U, ω) has an idempotent of weight 1, then (U, ω) is b-simple if, and only if, its only b-ideals are (0) and bar(U ). Let (U, ω) be a b-algebra. We define the b-radical of U, denoted by rad(U ), as: rad(U ) = (0), if (U, ω) is b-simple, otherwise as rad(U ) = T bar(B), where B runs over the maximal normal b-subalgebra of U. Of course, rad(U ) is a b-ideal of U.