The Hochschild-Mitchell dimension of linearly ordered sets and the continuum hypothesis (Q1920832)

As is known, the Hochschild-Mitchell dimension is \(n + 2\) for a totally ordered set with the supremum of cardinalities of closed intervals equal to \(\aleph_n\). On the other hand, an analogous claim, the general Mitchell conjecture [\textit{B. Mitchell}, Adv. Math. 8, 1-161 (1972; Zbl 0232.18009)] on the dimension of linearly ordered sets, is nor true for all linearly ordered sets if we do not require the equality \(2^{\aleph_n} = \aleph_{n + 1}\) for at least one natural \(n \geq 0\) [the author, Sib. Math. J. 33, No. 6, 1140-1143 (1992); translation from Sib. Math. Zh. 33, No. 6, 211-215 (1992; Zbl 0772.18008)]. Nevertheless, the Mitchell conjecture is corroborated for the subsets of the real line, but under the assumption \(2^{\aleph_0} = \aleph_1\) [the author, Sib. Math. J. 34, No. 4, 786-793 (1993; Zbl 0808.18010); translation from Sib. Mat. Zh. 34, No. 4, 217-227 (1993)]. Also, the answer is found to the Mitchell question about the dimension of the real line: \(\dim \mathbb{R} = 3\) independently of the continuum hypothesis [the author, Sov. Math., Dokl. 45, No. 1, 89-92 (1992; Zbl 0797.18009); translation from Dokl. Akad. Nauk SSSR 322, No. 2, 259-261 (1992)]. The aim of the present article is to prove equivalence of some claim on the Hochschild-Mitchell dimension of linearly ordered sets with the validity of the equalities \(2^{\aleph_n} = \aleph_{n + 1}\) for all natural \(n \geq 0\). Definition. We call a subset \(I \subseteq {\mathcal C}\) of a partially ordered set \({\mathcal C}\) semidense in \({\mathcal C}\) if, for all elements \(a < b\) in \({\mathcal C}\), there exists an \(i \in I\) such that \(a \leq i \leq b\). Given an arbitrary partially ordered set \({\mathcal C}\), we denote by \(\dim {\mathcal C}\) its Hochschild-Mitchell dimension. The central result of the article is the theorem: The following statements are equivalent: \((CH_\omega)\) for all natural \(n \geq 0\), the equality of cardinal numbers \(2^{\aleph_n} = \aleph_{n + 1}\) is valid; \((D)\) if a linearly ordered set \({\mathcal C}\) includes a semidense (in \({\mathcal C})\) subset of lesser cardinality that equals \(\aleph_n\) for a suitable \(0 \leq n < \infty\), then \(\dim {\mathcal C} = n + 3\).