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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
Full Text:
##### References:
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