$\alpha $-ideals in $0$-distributive posets
Khalid A. Mokbel · Mathematica Bohemica · 2015
The concept of $\alpha $-ideals in posets is introduced. Several properties of $\alpha $-ideals in $0$-distributive posets are studied. Characterization of prime ideals to be $\alpha $-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 $\alpha $-ideal. Moreover, it is shown that the set of all $\alpha $-ideals $\alpha \mathop {\rm Id}(P)$ 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 $\alpha $-ideals. Some counterexamples are also given.