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On posets with isomorphic interval posets

Authors:	Judita Lihová

Institution:	(1) Prírodovedecká fakulta UPJ, 041 54 Koice, Jesenná 5, Slovakia

Abstract:	Let $\mathbb{A} = (A, \leqslant )$ be a partially ordered set, Int $\mathbb{A}$ the system of all (nonempty) intervals of $\mathbb{A},$ partially ordered by the set-theoretical inclusion $\subseteq$ . We are interested in partially ordered sets $\mathbb{B} = (B, \leqslant )$ with Int $\mathbb{B}$ isomorphic to Int $\mathbb{A}$ . We are going to show that they correspond to couples of binary relations on A satisfying some conditions. If $\mathbb{A}$ is a directed partially ordered set, the only $\mathbb{B}$ with Int $\mathbb{B}$ isomorphic to Int $\mathbb{A}$ are $\mathbb{A}_1^\delta \times \mathbb{A}_2$ corresponding to direct decompositions $\mathbb{A}_1 \times \mathbb{A}_2$ of $\mathbb{A}$ ( $\mathbb{A}_1^\delta$ denotes the dual of $\mathbb{A}_1$ . The present results include those presented in the paper 11] by V. Slavík. Systems of intervals, particularly of lattices, have been investigated by many authors, cf. 1]–11].

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