On varieties of completely regular semigroups.
A language L ⊆A* is literally idempotent in case that ua2v ∈ L if and only if uav ∈ L, for each u,v ∈ A*, a ∈ A. Varieties of literally idempotent languages result naturally by taking all literally idempotent languages in a classical (positive) variety or by considering a certain closure operator on classes of languages. We initiate the systematic study of such varieties. Various classes of literally idempotent languages can be characterized using syntactic methods. A starting example is the...
A necessary and sufficient condition is given for a) a principal left ideal in to be equal to the direct product of the corresponding principal left ideals , b) an -class to be equal to the direct product of the corresponding -classes .