Geodesic
0-distributive posets
Mathematica Bohemica, Tome 138 (2013) no. 3, pp. 325-335

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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.
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.
DOI : 10.21136/MB.2013.143440
Classification : 06A06, 06A75, 06D75
Keywords: 0-distributive poset; ideal; semiprime ideal; prime ideal; semiatom; semiatomic 0-distributive poset 
Mokbel, Khalid A.; Kharat, Vilas S. 0-distributive posets. Mathematica Bohemica, Tome 138 (2013) no. 3, pp. 325-335. doi: 10.21136/MB.2013.143440
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