Mal'cev products of rings satisfying $x^n\approx x$ and idempotent semirings

Abstract

Let $\Rn$ be the variety of rings satisfying $x^n\eqid x$, where $n\geq2$, and let $\W$ be a variety of idempotent semirings with commutative addition. The equality $\Rn\circ\W=\Rn\vee\W$ holds, and this variety has subvariety lattice $L(\Rn)\times L(\W)$. Its subdirectly irreducible members that are not additively idempotent have exactly one nontrivial ring component, which is a finite field. Such a member is determined by this field and an idempotent semiring with a multiplicative identity; a separation condition on unary polynomial functions of the latter characterizes subdirect irreducibility. Every subvariety has a finite identity basis. A four-element semiring generates a variety with subdirectly irreducible members of unbounded cardinality. The results extend the Mal'cev product theorem of Wang and Shao from the absorption subvariety to arbitrary $\W\leq\Slp$.