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Article

MSC: 06A06, 06A75
Keywords:
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.
References:
[1] Beran, L.: Length of ideals in lattices. Collect. Math. 46 (1995), 21-33. MR 1366126 | Zbl 0842.06006
[2] Grätzer, G.: General Lattice Theory. New appendices by the author with B. A. Davey, et al., (Second ed.). Birkhäuser, Basel (1998). MR 1670580
[3] Grillet, P. A., Varlet, J. C.: Complementedness conditions in lattices. Bull. Soc. R. Sci. Liège 36 (1967), 628-642. MR 0228389 | Zbl 0157.34202
[4] Halaš, R.: Characterization of distributive sets by generalized annihilators. Arch. Math., Brno 30 (1994), 25-27. MR 1282110
[5] Halaš, R., Rachůnek, J.: Polars and prime ideals in ordered sets. Discuss. Math., Algebra Stoch. Methods 15 (1995), 43-59. MR 1369627
[6] Jayaram, C.: Semiatoms in semilattices. Math. Semin. Notes, Kobe Univ. 10 (1982), 351-366. MR 0704918 | Zbl 0522.06003
[7] Joshi, V. V., Waphare, B. N.: Characterizations of $0$-distributive posets. Math. Bohem. 130 (2005), 73-80. MR 2128360 | Zbl 1112.06001
[8] Kharat, V. S., Mokbel, K. A.: Primeness and semiprimeness in posets. Math. Bohem. 134 (2009), 19-30. MR 2504684 | Zbl 1212.06001
[9] Kharat, V. S., Mokbel, K. A.: Semiprime ideals and separation theorems for posets. Order 25 (2008), 195-210. DOI 10.1007/s11083-008-9087-3 | MR 2448404 | Zbl 1155.06003
[10] Pawar, Y. S.: 0-1 distributive lattices. Indian J. Pure Appl. Math. 24 (1993), 173-179. MR 1210389 | Zbl 0765.06015
[11] Pawar, Y. S., Dhamke, V. B.: $0$-distributive posets. Indian J. Pure Appl. Math. 20 (1989), 804-811. MR 1012883 | Zbl 0676.06009
[12] Pawar, Y. S., Thakare, N. K.: $0$-distributive semilattices. Can. Math. Bull. 21 (1978), 469-475. DOI 10.4153/CMB-1978-080-6 | MR 0523589
[13] Rachůnek, J.: On $0$-modular and $0$-distributive semilattices. Math. Slovaca 42 (1992), 3-13. MR 1159487
[14] Varlet, J. C.: A generalization of the notion of pseudo-complementedness. Bull. Soc. R. Sci. Liège 37 (1968), 149-158. MR 0228390 | Zbl 0162.03501
[15] Varlet, J. C.: Distributive semilattices and Boolean lattices. Bull. Soc. R. Sci. Liège 41 (1972), 5-10. MR 0307991 | Zbl 0237.06011
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