Several characterizations of 0-distributive posets are obtained by using the prime ideals as well as the semiprime ideals. It is also proved that if every proper $l$-filter of a poset is contained in a proper semiprime filter, then it is $0$-distributive. Further, the concept of a semiatom in 0-distributive posets is introduced and characterized in terms of dual atoms and also in terms of maximal annihilator. Moreover, semiatomic 0-distributive posets are defined and characterized. It is shown that...

In a 0-distributive lattice sufficient conditions for an $\alpha$-ideal to be an annihilator ideal and prime ideal to be an $\alpha$-ideal are given. Also it is proved that the images and the inverse images of $\alpha$-ideals are $\alpha$-ideals under annihilator preserving homomorphisms.