## 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:

 300000 Other combinatorial set theory 3e+35 Consistency and independence results

### Citations:

Zbl 0561.54004; Zbl 0718.54001
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