Power-associative algebras that are train algebras

We investigate the structure of power-associative algebras that are train algebras. We first show the existence of idempotents, which are all principal and absolutely primitive. We then study the train equation involving the Peirce decomposition. When the algebra is finite-dimensional, it turns out that the dimensions of the Peirce components are invariant and that the upper bounds for their nil-indexes are reached for some idempotent. Further, locally train algebras are shown to be train algebras. We then get a complete description of the set of idempotents by giving their explicit formulas, including several illustrative examples. Some attention is paid to the Jordan case, where we discuss conditions forcing power-associative train algebras to be Jordan algebras. It is also shown that finitely generated Jordan train algebras are finite-dimensional. For a nth-order Bernstein algebra of period p, we prove that power-associativity necessitates p=1. In this case, there are 2n−1 possible train equations, which are explicitly described.