PlanetMath: chain finite

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chain finite

(Definition)

A poset is said to be chain finite if every chain with both maximal and minimal element is finite.

$\mathbb{Z}$ with the standard order relation is chain finite, since any infinite subset of $\mathbb{Z}$ must be unbounded above or below. $\mathbb{Q}$ with the standard order relation is not chain finite, since for example $\{ x \in\nobreak \mathbb{Q} \,\mid\, 0 \leqslant\nobreak x \leqslant\nobreak 1 \}$ is infinite and has both a maximal element and a minimal element 0.

Chain finiteness is often used to draw conclusions about an order from information about its covering relation (or equivalently, from its Hasse diagram).

"chain finite" is owned by lars_h. [ owner history (2) ]

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chain finite

Keywords:

poset

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Cross-references: Hasse diagram, covering, conclusions, maximal element, infinite subset, order, finite, minimal element, chain, poset
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This is version 1 of chain finite, born on 2007-04-12.
Object id is 9180, canonical name is ChainFinite.
Accessed 787 times total.

Classification:

AMS MSC:

06A06 (Order, lattices, ordered algebraic structures :: Ordered sets :: Partial order, general)

Pending Errata and Addenda

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