On Quasi orthogonal Bernstein Jordan algebras

  • Ana Fuenzalida Departamento de Matemáticas, Universidad de Talca. Casilla 747. Talca, Chile.
  • Alicia Labra Departamento de Matemáticas, Universjdad de Chile. Casilla 653. Santiago, Chile.
  • Cristian Mallol Departamento de Mathématiques et lnformatique, Université de Montpellier III, B.P. 5043 34032 Montpellier, France.


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= KeUV 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.

How to Cite
A. Fuenzalida, A. Labra, and C. Mallol, “On Quasi orthogonal Bernstein Jordan algebras”, CUBO, no. 8, pp. 1-6, Dec. 1992.