Two operators on the lattice of completely regular semigroup varieties (Q1601812)

\textit{L. Polák} [Semigroup Forum 32, 97-123 (1985; Zbl 0564.20034)] introduced an operator \(\sim\to\overset\leftarrow\sim\) on the lattice of fully invariant congruences on the free unary semigroup on a countably infinite set of variables. In the paper under review this operator is further explored for fully invariant congruences \(\sim\) which contain the least completely regular congruence. The author proves that this operator gives rise to an idempotent complete endomorphism \(V\to\overset\leftarrow V\) of the lattice \(L(CR)\) of completely regular semigroup varieties. Let \(\rho_\ell\) be the complete congruence induced by the endomorphism \(V\to\overset\leftarrow V\) and \(\rho_r\) the complete congruence induced by the endomorphism \(V\to\overset\rightarrow V\) constructed in a dual way. It is shown that \(\rho_\ell\cap\rho_r\) is the equality on the principal filter of \(L(CR)\) generated by the variety of all semilattices. This theorem is not new but follows from the subdirect decomposition of \(L(CR)\) which is the subject of Theorem 15 of [\textit{F. Pastijn}, J. Aust. Math. Soc., Ser. A 49, No.~1, 24-42 (1990; Zbl 0706.20042)]. Other results of the author's Section 3 also immediately follow from the results of this paper. In the remaining sections the author investigates how the operators \(V\to\overset\leftarrow V\) and \(V\to\overset\rightarrow V\) interact with one another and with other similarly defined operators, special settings where the distributive law holds are established, and a solution of the word problem is given for the free objects in the variety of orthogroups for which the Green relation \(\mathcal H\) is a left congruence.