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Khalid A. Mokbel, Mathematics Department, Education Faculty, Hodaidah University, P. O. Box 3114, Al Hudaydah, Yemen, e-mail: khalidalaghbari@yahoo.com
Abstract: 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 $\alpha\mathop Id(P)$ 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.
Keywords: $0$-distributive poset; ideal; $\alpha$-ideal; prime ideal; non-dense ideal; minimal ideal; annihilator ideal
Classification (MSC 2010): 06A06, 06A75
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