Tying up baric algebras

Given two baric algebras (A 1, ω 1) and (A 2, ω 2) we describe a way to define a new baric algebra structure over the vector space A 1 ⊕ A 2, which we shall denote (A 1 ⋈ A 2, ω 1 ⋈ ω 2). We present some easy properties of this construction and we show that in the commutative and unital case it preserves indecomposability. Algebras of the form A 1 ⋈ A 2 in the associative, coutable-dimensional, zero-characteristic case are classified.

A note on train algebras

Train algebras were first introduced by Etherington in (1) and proved very useful in dealing with problems in mathematical genetics. The types of algebras which arose were commutative, non-associative and finite-dimensional. It proved convenient in the general theory to regard them as defined over the complex numbers. We remind the reader of some basic definitions. A baric algebra is one which admits a non-trivial homomorphism into its coefficient field K. A (principal) train algebra is baric and has a rank equation in which the coefficients of a general element x depend only on its baric value, generally called the weight of x. A special train algebra (STA) is a baric algebra in which the nilideal is nilpotent and all its right powers are ideals; the nilideal being the set of elements of A of weight zero. In (2) Etherington showed that in a baric algebra one can always take a very simple basis consisting of a distinguished element of unit weight and all other basis elements of weight zero.