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 a 0-distributive poset P is semiatomic if and only if the intersection of all non dense prime ideals of P equals (0]. Some counterexamples are also given.
The concept of α-ideals in posets is introduced. Several properties of α-ideals in 0-distributive posets are studied. Characterization of prime ideals to be α-ideals in 0- distributive posets is obtained in terms of minimality of ideals. Further, it is proved that if a prime ideal I of a 0-distributive poset is non-dense, then I is an α-ideal. Moreover, it is shown that the set of all α-ideals α Id(P) of a poset P with 0 forms a complete lattice. A result analogous to separation theorem for finite 0-distributive posets is obtained with respect to prime α-ideals. Some counterexamples are also given.