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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)

MSC:
06A06 Partial orders, general
Keywords:
poset; interval; lattice
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References:
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