Hausdorff’s theorem for posets that satisfy the finite antichain property.

*(English)*Zbl 0934.06005A poset \(P\) is well-founded iff it has no infinite decreasing sequence; \(P\) satisfies the condition FAC iff every antichain of \(P\) is finite; \(P\) is scattered iff it has no subset which has the order-type of the set of rational numbers with their natural order. F. Hausdorff [Grundzüge der Megenlehre (Veit & Comp., Leipzig) (1914; JFM 45.0123.01), Kapitel IV, §6] had considered this concept only for linearly ordered sets. If \(P\) and \(Q\) are posets with the same carrier sets and orders \(\leq_P\) resp. \(\leq_Q\), then \(Q\) augments \(P\), iff \(\leq_Q\) contains \(\leq_P\). The set of all possible augmentations of \(P\) is denoted by \(\text{aug}(P)\).

When \(P\) satisfies the FAC, \(({\mathcal A}(P),\supset)\) is the poset of all nonempty antichains of \(P\) under inverse inclusion. This set is well-founded and thus it has a rank function \(\text{rk}_{\mathcal A}\). The antichain rank of \(P\) is then its image \(\text{rk}_{\mathcal A}(P)\), which is an ordinal. The main theorem then states the following: Let \(\rho\) be an ordinal and let \(\text{aug} ({\mathcal H}^\rho)\) be the closure of the class of all well-founded posets with antichain rank \(\leq \rho\) under inversion, lexicographic sums, and augmentation. Then it contains the class of all scattered FAC-posets with rank \(\leq\rho\). So \(\text{aug}({\mathcal H})\), which is the closure of the well-founded posets with FAC under inversion, lexicographic sums, and augmentation, is the class of all scattered FAC-sets. For \(\rho=1\) this implies Hausdorff’s theorem [loc. cit.]. In the proof, the authors introduce a new product operation for ordinals, which they call the Hessenberg based product.

When \(P\) satisfies the FAC, \(({\mathcal A}(P),\supset)\) is the poset of all nonempty antichains of \(P\) under inverse inclusion. This set is well-founded and thus it has a rank function \(\text{rk}_{\mathcal A}\). The antichain rank of \(P\) is then its image \(\text{rk}_{\mathcal A}(P)\), which is an ordinal. The main theorem then states the following: Let \(\rho\) be an ordinal and let \(\text{aug} ({\mathcal H}^\rho)\) be the closure of the class of all well-founded posets with antichain rank \(\leq \rho\) under inversion, lexicographic sums, and augmentation. Then it contains the class of all scattered FAC-posets with rank \(\leq\rho\). So \(\text{aug}({\mathcal H})\), which is the closure of the well-founded posets with FAC under inversion, lexicographic sums, and augmentation, is the class of all scattered FAC-sets. For \(\rho=1\) this implies Hausdorff’s theorem [loc. cit.]. In the proof, the authors introduce a new product operation for ordinals, which they call the Hessenberg based product.

Reviewer: E.Harzheim (Köln)