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Cardinal invariants of the lattice of partitions. (English) Zbl 1034.03050

There is a natural order on the set of all partitions of \(\omega \), namely, \(X\leq Y\) if each member of \(X\) is a union of some members of \(Y\). Attempting to mimick the analogy between the orders \(\subseteq \) and \(\subseteq ^*\) (containment modulo a finite set) on subsets of \(\omega \), the author introduces four new orders on the set of all partitions of \(\omega \): \(\leq _1^*\), \(\leq _2^*\) and their inverses \(\preceq _1^*\), \(\preceq _2^*\). The definitions are quite natural. For two partitions \(X\) and \(Y\) of \(\omega \), \(X\leq _1^*Y\) if each member of \(X\), except finitely many, is a union of some members of \(Y\), and \(X\leq _2^* Y\), if for some finite set \(d\) and each \(x\in X\), \(y\in Y\), either \(y\setminus x\subseteq d\) or \(y\cap x\subseteq d\). It turns out that all four orders allow us to define cardinal invariants analogous to \(\mathfrak p\), \(\mathfrak t\), \(\mathfrak h\), \(\mathfrak s\), \(\mathfrak r\) and \(\mathfrak a\) from E. K. van Douwen’s paper [Handbook of set-theoretical topology, 111–167 (1984; Zbl 0561.54004)] or J. Vaughan’s paper [in: Open problems in topology, 195–218 (1990; Zbl 0718.54001)]. The aim of the present paper is to investigate these analogous cardinal invariants. It turns out that a full analogy of a diagram of inequalities holds for \(\preceq _1^*\), and MA or Cohen reals have the well-known effects, too. The case of the remaining three orders is slightly different and the paper gives a complete description of it.
Reviewer: Petr Simon (Praha)

MSC:

03E05 Other combinatorial set theory
03E35 Consistency and independence results
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