On the Pin-Thérien expansion of completely regular monoids (Q1961690)

Motivated by a problem in formal language theory, \textit{J.-E.~Pin} and \textit{D.~Thérien} [Int. J. Algebra Comput. 3, No. 4, 535-555 (1993; Zbl 0816.20065)] introduced, for each finite monoid, a certain expansion. For a (pseudo)variety \(\mathbf V\) of finite monoids, let \(\overline{\mathbf V}\) denote the smallest variety containing \(\mathbf V\) which is closed under forming the Pin-Thérien expansions of its members. \textit{M.~J.~J.~Branco} [Semigroup Forum 49, No. 3, 329-334 (1994; Zbl 0821.20043)] described the varieties \(\overline{\mathbf V}\) for all varieties \(\mathbf V\) of finite idempotent monoids. In the present note, the author observes that Branco's result easily extends to varieties of finite completely regular monoids. Namely, if \(\mathbf V\) is such a variety, let \(\mathbf{RV}\) denote the class of all finite monoids whose regular elements form a submonoid which lies in \(\mathbf V\) and \(\mathbf W\) the class of all finite monoids satisfying the pseudoidentities \(x^\omega(xy)^\omega=(xy)^\omega=(xy)^\omega y^\omega\). Then \(\overline{\mathbf V}=\mathbf{RV}\cap{\mathbf W}\).