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Partially ordered sets having selfdual system of intervals. (English) Zbl 0934.06004

If \(P\) is a partially ordered set, denote by Int\(_0P\) the system of all intervals of \(P\) including the empty set and by \(\text{Int }P= \text{Int}_0P\setminus \{\emptyset \}\), both ordered by the set-theoretical inclusion. Let \(S_0\) (resp. \(S\)) be the class of all partially ordered sets such that Int\(_0P\) (resp. \(\text{Int }P\)) is selfdual. It is known, see J. Jakubík [Czech. Math. J. 41, 135-140 (1991; Zbl 0790.06001)], that if \(P\in S_0\) then \(P\) is a lattice and \(\text{card }P \leq 4\). In the paper, the author proves, contrary to \(S_0\), that for any infinite cardinal \(\alpha \) there exists a connected homogeneous partially ordered set \(P_{\alpha }\) such that \(P_{\alpha } \in S\) and \(\text{card }P_{\alpha } = \alpha \).

MSC:

06A06 Partial orders, general
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