On Quasi orthogonal Bernstein Jordan algebras

Abstract

Bernstein algebras were introduced by P. Holgate in [1] to deal with the problem of populations which are in equilibrium after the second generation. In [3] we work with Weak Bernstein Jordan algebras, i.e. a class of commutative algebras with idempotent element and defined by relations. In [3, section 4] we prove that if A= Ke ⊕ U ⊕ V is the Pierce decomposition of A relative to the idempotent e, then the situations U³ = {0} and U²(UV) = {0} are independents of the different Pierce decompositions of A, then they are invariants of A. We say that A is orthogonal if U³ = {0} and quasiorthogonal if U²(UV) = {0}. The orthogonality case was treated in [2].

In this paper we prove that every Bernstein-Jordan algebra of dimension less than 11 is quasi-orthogonal. Moreover we prove that there exists only one non quasi-orthogonal Bernstein-Jordan algebra of dimension 11.