On a semigroup induced by a distributive lattice (Q749678)

The Popov semigroup S(R) of a distributive lattice R was introduced in \textit{Yu. F. Popov} [Semigroup Forum 33, 145-148 (1986; Zbl 0582.06011)]. The Popov semigroups satisfy: (1) S(R) is a left regular band (i.e. band satisfying \(xyx=xy)\). (2) Green's relation \({\mathcal R}\) is diagonal and \({\mathcal L}={\mathcal D}={\mathcal J}\) is a congruence. In this paper the authors prove that moreover: (3) S(R)/\({\mathcal L}\) is up- directed (as a meet semilattice). (4) There exists a semilattice ideal T of S(R)/\({\mathcal L}\) which is a retract of S(R)/\({\mathcal L}\). (5) T is a distributive lattice isomorphic to R. An example is given to show that conditions (1)-(5) are not sufficient for a semigroup to be a Popov semigroup for some distributive lattice. It is proved that if S is a regular band then S/\({\mathcal L}\) is isomorphic to a subsemigroup of a Popov semigroup S(R) for some distributive lattice R.