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On posets with isomorphic interval posets. (English) Zbl 0953.06002
Let \((A,\leq)\) be a partially ordered set (poset). By an interval of \(A\) is meant a nonempty set \(\{x\in A; a\leq x \leq b\}\), for some \(a,b\in A\), \(a\leq b\). Denote by \(\operatorname {Int} A\) the poset of all intervals of \(A\) ordered by set inclusion. It is proved that a directed poset \(A\) and a poset \(B\) have isomorphic posets \(\operatorname {Int} A\) and \(\operatorname {Int} B\) if and only if there exist posets \(C\), \(D\) such that \(A\) is isomorphic to \(C\times D\) and \(B\) is isomorphic to \(C^*\times D\). Here \(C^*\) denotes the poset dually isomorphic to \(C\).
Reviewer: V.Slavík (Praha)

06A06 Partial orders, general
poset; interval; lattice
Full Text: DOI EuDML
[1] V. I. Igošin: Selfduality of lattices of intervals of finite lattices. Inst. Matem. Sibir. Otdel. AN SSSR, Meždunarodnaja konferencija po algebre posvjaščennaja pamjati A. I. Maľceva, Tezisy dokladov po teoriji modelej i algebraičeskich sistem, Novosibirsk 1989, p. 48.
[2] V. I. Igošin: Lattices of intervals and lattices of convex sublattices of lattices. Uporjadočennyje množestva i rešotki. Saratov 6 (1990), 69-76.
[3] V. I. Igošin: Identities in interval lattices of lattices. Coll. Math. Soc. J. Bolyai 33 (Contributions to Lattice Theory), Szeged 1980 (1983), 491-501.
[4] V. I. Igošin: On lattices with restriction on their intervals. Coll. Math. Soc. J. Bolyai 43 (Lectures in Universal Algebra), Szeged 1983 (1986), 209-216.
[5] V. I. Igošin: Algebraic characteristic of lattices of intervals. Uspechi matem. nauk 40 (1985), 205-206.
[6] V. I. Igošin: Semimodularity in lattices of intervals. Math. Slovaca 38 (1988), 305-308. · Zbl 0664.06007
[7] J. Jakubík: Selfduality of the system of intervals of a partially ordered set. Czechoslov. Math. J. 41 (1991), 135-140. · Zbl 0790.06001
[8] J. Jakubík, J. Lihová: Systems of intervals of partially ordered sets. Math. Slovaca, to appear. · Zbl 0888.06001
[9] M. Kolibiar: Intervals, convex sublattices and subdirect representations of lattices. Universal Algebra and Applications, Banach Center Publications, Vol. 9, Warsaw 1982, 335-339. · Zbl 0506.06003
[10] J. Lihová: Posets having a selfdual interval poset. Czechoslovak Math. J. 44 (1994), 523-533. · Zbl 0822.06001
[11] V. Slavík: On lattices with isomorphic interval lattices. Czechoslovak Math. J. 35 (1985), 550-554. · Zbl 0592.06003
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