Some remarks on a problem of J. D. Monk (Q1805108)

The reviewer stated the following problem in 1983 [Period. Math. Hung. 14, 269-308 (1983; Zbl 0517.03027)]: (P) If \(A_ \alpha\) is an interval Boolean algebra for each \(\alpha< \kappa\), is the independence of \(\prod_{\alpha< \kappa} A_ \alpha\) equal to \(2^ \kappa\)? The first author recently answered (P) positively [Math. Jap. 39, 1-5 (1994; Zbl 0805.06012)]. The present paper is concerned with some variants of (P), formulated in terms just of orderings. Some of the results are superseded by the result of the first author [loc. cit.]. But Theorem 3.1 remains of interest: If ZF is consistent then so is ZFC+GCH+ ``there are orderings \(\langle \prec_ n: n<\omega \rangle\) on \(\omega_ 2\) with the property that \(\forall \alpha< \beta< \gamma< \omega_ 2 \exists n\in \omega (\gamma\in (\alpha, \beta)_{\prec_ n})\)''. This is a result concerning the representation of orderings, a notion of A. Hajnal: let \(\Sigma\) be a set of orderings on a fixed natural number \(n\). Then \(\Sigma\) is \((\kappa, \lambda)\)-representable if there is a set \({\mathcal L}\) of orderings on \(\kappa\) with \(|{\mathcal L}|= \lambda\) such that for each \(\alpha_ 0< \alpha_ 1< \dots< \alpha_{n-1}< \kappa\) there exist \(\ll\in \Sigma\) and \(\prec\in {\mathcal L}\) such that \(i\ll j\) iff \(\alpha_ i \prec \alpha_ j\) for any \(i<j <n\). Hajnal raised the problem, still open, to characterize the representable sets of orderings.