Plenary train algebras of rank m and backcrossing identity.

This paper concerns commutative plenary train algebras of rank m and their idempotents. We obtain the Peirce decomposition of these algebras having an idempotent element and the multiplication table of Peirce components when the plenary train roots are mutually different. We show that a backcrossing algebra is a plenary train algebra of rank m if and only if, it is a principal train one of rank m. For the backcrossing train algebras, we confirm a first conjecture of Juan Carlos Gutiérrez Fernández on the relation between plenary train roots and principal train roots. A second conjecture on the existence of idempotent in train algebras also obtains a positive answer in the class of backcrossing algebras. [ABSTRACT FROM AUTHOR]