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$0$-distributive poset; ideal; $\alpha $-ideal; prime ideal; non-dense ideal; minimal ideal; annihilator ideal
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.
[1] Balasubramani, P., Venkatanarasimhan, P. V.: Characterizations of the $0$-distributive lattice. Indian J. Pure Appl. Math. 32 (2001), 315-324. MR 1826759 | Zbl 0984.06007
[2] Cornish, W. H.: Annulets and $\alpha$-ideals in a distributive lattice. J. Aust. Math. Soc. 15 (1973), 70-77. DOI 10.1017/S1446788700012775 | MR 0344170 | Zbl 0274.06008
[3] Grätzer, G.: General Lattice Theory. New appendices by the author with B. A. Davey et al. Birkhäuser Basel (1998). MR 1670580
[4] Grillet, P. A., Varlet, J. C.: Complementedness conditions in lattices. Bull. Soc. R. Sci. Liège (electronic only) 36 (1967), 628-642. MR 0228389 | Zbl 0157.34202
[5] Halaš, R.: Characterization of distributive sets by generalized annihilators. Arch. Math., Brno 30 (1994), 25-27. MR 1282110
[6] Halaš, R., Rachůnek, J.: Polars and prime ideals in ordered sets. Discuss. Math., Algebra Stoch. Methods 15 (1995), 43-59. MR 1369627
[7] Jayaram, C.: Prime {$\alpha$}-ideals in an {$0$}-distributive lattice. Indian J. Pure Appl. Math. 17 (1986), 331-337. MR 0835346
[8] Joshi, V. V., Mundlik, N.: Prime ideals in $0$-distributive posets. Cent. Eur. J. Math. 11 (2013), 940-955. MR 3032342 | Zbl 1288.06002
[9] Joshi, V. V., Waphare, B. N.: Characterizations of $0$-distributive posets. Math. Bohem. 130 (2005), 73-80. MR 2128360 | Zbl 1112.06001
[10] 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
[11] Pawar, Y. S., Khopade, S. S.: $\alpha$-ideals and annihilator ideals in $0$-distributive lattices. Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 49 (2010), 63-74. MR 2797524 | Zbl 1245.06023
[12] Pawar, Y. S., Mane, D. N.: $\alpha$-ideals in $0$-distributive semilattices and $0$-distributive lattices. Indian J. Pure Appl. Math. 24 (1993), 435-443. MR 1234802 | Zbl 0789.06005
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