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$0$-distributive poset; $0$-ideal; $\alpha $-ideal; prime ideal; semiprime ideal; dense ideal

References:

[1] 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

[2] Cornish, W. H.: **$0$-ideals, congruences, and sheaf representations of distributive lattices**. Rev. Roum. Math. Pures Appl. 22 (1977), 1059-1067. MR 0460202 | Zbl 0382.06011

[3] Grätzer, G.: **General Lattice Theory**. Birkhäuser, Basel (1998). MR 1670580

[4] Halaš, R.: **Characterization of distributive sets by generalized annihilators**. Arch. Math., Brno 30 (1994), 25-27. MR 1282110

[5] Halaš, R., Rachůnek, R. J.: **Polars and prime ideals in ordered sets**. Discuss. Math., Algebra Stoch. Methods 15 (1995), 43-59. MR 1369627

[6] Jayaram, C.: **$0$-ideals in semilattices**. Math. Semin. Notes, Kobe Univ. 8 (1980), 309-319. MR 0601900

[7] Jayaram, C.: **Quasicomplemented semilattices**. Acta Math. Acad. Sci. Hung. 39 (1982), 39-47. DOI 10.1007/BF01895211 | MR 0653670 | Zbl 0516.06002

[8] Joshi, V. V., Waphare, B. N.: **Characterizations of $0$-distributive posets**. Math. Bohem. 130 (2005), 73-80. MR 2128360 | Zbl 1112.06001

[9] Kharat, V. S., Mokbel, K. A.: **Primeness and semiprimeness in posets**. Math. Bohem. 134 (2009), 19-30. MR 2504684 | Zbl 1212.06001

[10] Mokbel, K. A.: **$\alpha$-ideals in $0$-distributive posets**. Math. Bohem. (2015), 140 319-328. MR 3397260 | Zbl 1349.06001