Complete reducibility of systems of equations with respect to R

Abstract

It is shown that the pseudovariety $\mathsf{R}$ of all finite $\mathcal{R}$-trivial semigroups is completely reducible with respect to the canonical signature. Informally, if the variables in a finite system of equations with rational constraints may be evaluated by pseudowords so that each value belongs to the closure of the corresponding rational constraint and the system is verified in $\mathsf{R}$, then there is some such evaluation which is “regular”, that is one in which, additionally, the pseudowords only involve multiplications and $\omega$-powers.