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# Article

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Keywords:
$0$-distributive poset; $0$-ideal; $\alpha$-ideal; prime ideal; semiprime ideal; dense ideal
Summary:
The concept of a $0$-ideal in \mbox {$0$-distributive} posets is introduced. Several properties of $0$-ideals in $0$-distributive posets are established. Further, the interrelationships between $0$-ideals and \mbox {$\alpha$-ideals} in \mbox {$0$-distributive} posets are investigated. Moreover, a characterization of prime ideals to be $0$-ideals in $0$-distributive posets is obtained in terms of non-dense ideals. It is shown that every $0$-ideal of a $0$-distributive meet semilattice is semiprime. Several counterexamples are discussed.
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

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