chain finite

A poset is said to be chain finite if every chain with bothmaximal (http://planetmath.org/MaximalElement) and minimal element is finite.

$\\mathbb{Z}$ with the standard order relation is chain finite,since any infinite subset of $\\mathbb{Z}$ must beunbounded (http://planetmath.org/UpperBound) above or below. $\\mathbb{Q}$ with the standard order relation is not chain finite,since for example $\\{x\\in\obreak\\mathbb{Q}\\,\\mid\\,0\\leqslant\obreak x\\leqslant\obreak 1\\}$ is infinite and has both a maximal element $1$ 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).

单词	ChainFinite
释义	chain finite A poset is said to be chain finite if every chain with bothmaximal (http://planetmath.org/MaximalElement) and minimal element is finite. $\\mathbb{Z}$ with the standard order relation is chain finite,since any infinite subset of $\\mathbb{Z}$ must beunbounded (http://planetmath.org/UpperBound) above or below. $\\mathbb{Q}$ with the standard order relation is not chain finite,since for example $\\{x\\in\obreak\\mathbb{Q}\\,\\mid\\,0\\leqslant\obreak x\\leqslant\obreak 1\\}$ is infinite and has both a maximal element $1$ 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).
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