Notes on mildly distributive semilattices

In this paper we shall investigate the mildly distributive meet-semilattices by means of the study of their filters and Frink-ideals as well as applying the theory of annihilator. We recall some characterizations of the condition of mildly-distributivity and we give several new characterizations. We prove that the definition of strong free distributive extension, introduced by Hickman in 1984, can be simplified and we show a correspondence between (prime) Frink-ideals of a mildly distributive semilattice and (prime) ideals of its strong free distributive extension.