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Selfduality of the system of intervals of a partially ordered set. (English) Zbl 0790.06001

[See also the corrections announced below.]
For a partially ordered set \(P\) we denote by \(\text{Int }P\) the system of all intervals \([a,b]= \{x\in P: a\leq x\leq b\}\), where \(a,b\in P\) and \(a\leq b\), including the empty set. The system \(\text{Int }P\) is partially ordered by the set-theoretical inclusion. If \(P\) is a lattice, then \(\text{Int }P\) is a lattice as well. In general, \(\text{Int }P\) need not be a lattice.
V. I. Igoshin presented the following theorem: (A) Let \(L\) be a finite lattice. Then \(\text{Int }L\) is selfdual if and only if either (i) card \(L\leq 2\) or (ii) card \(L=4\) and \(L\) has two atoms.
Next, Igoshin proposed the problem whether there exists an infinite lattice \(L\) such that \(\text{Int } L\) is selfdual.
In the present paper it will be shown that the answer to this problem is negative. Namely, the following result will be proved: (B) Let \(P\) be a partially ordered set with card \(P>4\). Then the partially ordered system \(\text{Int }P\) is not selfdual.

MSC:

06A06 Partial orders, general
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References:

[1] В. И. Игошин: Самодвойственность решеток интервалов конечных решеток. Инст. матем. Сибир. Отдел. АН СССР, Международная конференция по алгебре посвященная памяти А. И. Мальцева, Тезисы докладов по теории моделей и алгебраических систем, Новосибирск 1989, c. 48. · Zbl 1170.01311
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[9] V. Slavík: On lattices with isomorphic interval lattices. Czechoslov. Math. J. 35, 1985, 550-554. · Zbl 0592.06003
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