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Stone ordered set; prime ideal; distributive pseudocomplemented ordered set; $l$-ideal
A distributive pseudocomplemented set $S$ [2] is called Stone if for all $a\in S$ the condition $LU(a^*,a^{**})=S$ holds. It is shown that in a finite case $S$ is Stone iff the join of all distinct minimal prime ideals of $S$ is equal to $S$.
[1] Grätzer G., Schmidt E. T.: On a problem of M.H. Stone. Acta Math. Sci. Hungarica 8 (1957), 455-460. DOI 10.1007/BF02020328 | MR 0092763
[2] Halaš R.: Pseudocomplemented ordered sets. Arch. Math. (Brno) 29 (1993), no. 3-4, 153-160. MR 1263116
[3] Halaš R.: Ideals, polars and annihilators in ordered sets. PҺD thesis, Olomouc, 1994.
[4] Chajda I.: Complemented ordered sets. Aгch. Math. (Brno) 28 (1992), 25-34. MR 1201863 | Zbl 0785.06002
[5] Chajda I., Rachůnek J.: Forbidden configurations for distributive and modular ordered sets. Order 5 (1989), 407-423. DOI 10.1007/BF00353659 | MR 1010389
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