The bar-radical of baric algebras

Abstract

1. Baric algebras. Baric algebras play a central role in the theory of genetic algebras. They were introduced by I. M. H. Etherington, [5], aiming for an algebraic treatment of Genetic Populations. Several classes of baric algebras have been defined: train, Bernstein, special triangular, etc. Let F be a field, A an algebra over F, not necessarily associative, commutative or finite dimensional. If co: A ~ F is a nonzero homomorphism, then the ordered pair (A, co) will be called a baric algebra over F and co its weight function. For x e A, co (x) is called the weight of x. If B~A, we will denote bar(B)= {xeB:co(x)=O}. We observe that bar (B) = B if and only if B c= bar (A). When B is a subalgebra of A and B $ bar (A), then B is called baric subalgebra of (A, og). In this case (B, co') is a baric algebra, where co' = colB: B ~ F. If B is a baric subalgebra of (A, co) and bar(B) is a two-sided ideal of bar (A) (then by [6, Prop. 1.1] it is also a two-sided ideal of A), then B is called normal baric subalgebra of(A, co). In the first case we will denote B < A or B < A ifB 4= A, in the second case B~ A or B<~ A if B r A. Let B be a baric subalgebra of (A, co). Then bar(B) is a two-sided ideal of B of codimension 1, called bar ideal of B. For all b e B with co (b) r 0, we have B = F b | bar (B). If I ~ bar (B) is a two-sided ideal of B, then I is called baric ideal of B. A baric homomorphism from (A, co) to (A', o9') is a homomorphism of F-algebras cp: A --. A' such that co' o cp = co. We define baric epimorphism, etc., of natural manner. If q~ and p' are baric homomorphisms and ~o o qr is defined, then it is also a baric homomorphism. If (o is bijective, then ~o- 1 is also baric. We wilt use (A, co) -~b ( A~, co') or A -~b A', to denote the existence of a baric isomorphism from (A, co) to (A', co'). For details about baric algebras the reader is referred to [91 and [12]. It is known that, if A and B are F-algebras, C c= A and D __ B are subalgebras and q~: A ~ B is a homomorphism, then (p- ~ (~o (C)) = C if ker ~0 _c C and ~0 (cp- 1 (D)) = D if ~p is an epimorphism.