Train algebras of degree 2 and exponent 3

Algebras Satisfying Polynomial Identity of Degree Six that are Principal Train

“…Let (A, ω) be a baric commutative K-algebra not necessarily associative verifying the identity (1). The partial linearisation of this identity gives the following result: Proposition 1.…”

“…Following him, several authors studied various classes of algebras satisfying polynomial identities, in order to model the process of genetic transmission. (see, for instances, [9], [1], [2]). The aim of this paper is to study the algebras verifying the polynomial identity 2x 2 x 4 = ω(x) 2 x 4 + ω(x) 4 x 2 that are principal train algebras.…”

Evolution algebras satisfying train identity of degree 2 and exponent 3

“…The Peirce decomposition of A is given by A = Ke ⊕ A 0 (e) ⊕ A 1/2 (e) ⊕ A −1/2 (e), where e is a nonzero idempotent and A λ = {x ∈ ker ω | ex = λx} for λ ∈ {0, 1/2, −1/2}. The relations between the Peirce subspaces are given by: [2,Theorem 3.4].…”

“…Example 3.7. Let (A, ω) be a K-baric evolution algebra of dimension 5 whose table of multiplication in the natural basis B = (e 1 , e 2 , e 3 , e 4 , e 5 ) is given by: e 2 1 = e 1 + e 2 + e 4 , e ;…”

“…These algebras were defined to model Hardy Weinberg's principle which states that the genetic heritage of a population stabilizes in the second generation. In [2] and [15], the authors studied the algebras verifying a train identity of degree 2 and exponent n for n = 3 or 4, that is, such that (x n ) 2 = ω(x) n x n (n = 3 or n = 4). They showed that this last class of algebras contains strictly that of Bernstein algebras.…”