Polák Theorem

One of the principal results of the study of varieties of completely regular semigroups is the construction of an isomorphic copy of the lattice \([\mathcal {S}, \mathcal{C}\mathcal{R}]\) in terms of functions between certain sets satisfying mostly natural conditions. Even though this construction is quite involved, as it necessarily must be if it is to describe such a complicated lattice, it fortunately turns out to be remarkably useful in its various applications. These include diverse global properties of the interval \([\mathcal {S}, \mathcal{C}\mathcal{R}]\) as well as detailed descriptions of certain of its parts. The proof of this theorem involves somewhat lengthy considerations and repeated applications of many concepts introduced and results obtained heretofore especially in Chap. 2.