## 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

### Keywords:

partially ordered set; interval; selfduality

### Citations:

Zbl 0790.06001; Zbl 0790.06002
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