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Keywords:
0-distributive poset; ideal; semiprime ideal; prime ideal; semiatom; semiatomic 0-distributive poset
0-distributive poset; ideal; semiprime ideal; prime ideal; semiatom; semiatomic 0-distributive poset
Summary:
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.
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