Algebraic structure of plenary train algebras of rank 4

In this paper, we study plenary train algebras of rank 4 admitting an idempotent. We obtain the Peirce decomposition of these algebras relative to an idempotent and the product of the associated Peirce components. The effect of changing an idempotent is discussed under the e-stability hypothesis. We show that a back-crossing algebra is a plenary train algebra of rank 4 if and only if it is a principal train algebra of rank 4. Finally, we study the particular case of monogenic back-crossing train algebras.