Dibaric Algebras

Abstract

Here we give basic properties of dibaric algebras which are motivated by genetic models. Dibaric algebras are not associative and they have a non trivial homomorphism onto the sex differentiation algebra. We define first join of dibaric algebras next indecomposable dibaric algebras. Finally, we prove the uniqueness of the decomposition of a dibaric algebra, with semiprincipal idempotent, as the join of indecomposable dibaric algebras.

Gonosomal algebra

“…Let θ and θ respectively the recombination rates between these two loci in females and males. During meiosis a female (a i b j , a k b l ) produces ova in the following proportions: 1…”

“…Algebraically, given three spaces A, B and C with respective bases (a 1 , a 2 ), (b 1 , b 2 ) and (c 1 , c 2 ) and with the algebra law: xy = 1 2 x + 1 2 y, the space A ⊗ B ⊗ C is equipped with the algebraic structure (a i ⊗ b j ⊗ c k ) (a p ⊗ b q ⊗ c r ) = (a i a p ) ⊗ (b j b q ) ⊗ (c k c r ) and the weight function ω (a i ⊗ b j ⊗ c k ) = 1. For i, j, k ∈ {1, 2} we note e (i, j,k) = a i ⊗ b j ⊗ c k , and we put: =    e (1,1,1) ⊗ e (1,1,1) , e (1,1,1) ⊗ e (2,1,1) , e (1,1,1) ⊗ e (2,2,1) , e (1,2,1) ⊗ e (2,2,1) , e (1,1,1) ⊗ e (2,1,2) , e (1,1,1) ⊗ e (2,2,2) , e (1,2,1) ⊗ e (2,2,2) , e (1,1,2) ⊗ e (2,1,2) , e (1,1,2) ⊗ e (2,2,2) , e (1,2,2) ⊗ e (2,2,2)    = e (1,1,1) ⊗ e (1,1,2) , e (1,1,2) ⊗ e (1,1,2) , e (1,1,1) ⊗ e (1,2,1) , e (1,2,1) ⊗ e (1,2,1) , e (1,1,1) ⊗ e (1,2,2) , e (1,2,1) ⊗ e (1,2,2) , e (1,1,2) ⊗ e…”

“…If A is a dibaric algebra then A 2 is baric. There are other results concerning dibaric algebras in [1] and [13].…”

Dibaric and evolution algebras in biology

“…The algebras A in which (3.9) holds are called stationary or Bernstein algebras because of the connection between this class and the problem of Bernstein (see [13,Section 2.1 and Chapters 4,5]). For an evolution algebra B, condition (3.9) is equivalent to the stationary principle…”

“…In [5] basic properties of dibaric algebras are given. The authors define the union of two dibaric algebras, following the same lines that were used by Costa and Guzzo [4] for baric algebras having an idempotent element of weight 1, and also the notion of indecomposable algebra and the results obtained for baric algebras are generalized for dibaric algebras.…”