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Linear extension
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Linear extension : ウィキペディア英語版
Linear extension
In order theory, a branch of mathematics, a linear extension of a partial order is a linear order (or total order) that is compatible with the partial order. As a classic example, the lexicographic order of totally ordered sets is a linear extension of their product order.
==Definitions==
Given any partial orders ≤ and ≤
*
on a set ''X'', ≤
*
is a linear extension of ≤ exactly when (1) ≤
*
is a linear order and (2) for every ''x'' and ''y'' in ''X'', if , then . It is that second property that leads mathematicians to describe ≤
*
as extending ≤.
Alternatively, a linear extension may be viewed as an order-preserving bijection from a partially ordered set ''P'' to a chain ''C'' on the same ground set.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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