Abstract
The aim of this paper is the study of relations between some special classes of baric algebras, as Jordan algebras, Bernstein or second order Bernstein algebras, train algebras and power-associative algebras. It will be proved that the train equation of a Bernstein algebra that is train of rank r is unique, what leads to a characterization of these algebras similar to the well known characterization of a Jordan-Bernstein algebra in terms of a Peirce decomposition.
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