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Hausdorff’s theorem for posets that satisfy the finite antichain property. (English) Zbl 0934.06005
A 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.
Reviewer: E.Harzheim (Köln)

MSC:
 06A06 Partial orders, general 03E10 Ordinal and cardinal numbers
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