Linear extension of a poset

Linear extension of a poset --- линейное расширение чу-множества.

Given a poset [math]\displaystyle{ P = (X, \leq) }[/math], a linear extension of [math]\displaystyle{ P }[/math] is a poset [math]\displaystyle{ L = (X, \leq') }[/math] [note, that ground sets are the same] with the properties that (a) [math]\displaystyle{ L }[/math] is a linear order and (b) for all [math]\displaystyle{ x,y \in X }[/math], if [math]\displaystyle{ x \leq y }[/math], then [math]\displaystyle{ x \leq' y }[/math] [extension]. A family [math]\displaystyle{ {\mathcal R} = \{L_{1}, L_{2}, \ldots, L_{t}\} }[/math] of linear extensions of [math]\displaystyle{ P = (X,\leq) }[/math] is called a realizer of [math]\displaystyle{ P }[/math] provided that, for all [math]\displaystyle{ x,y \in X, \; x \leq y }[/math] if and only if [math]\displaystyle{ x \leq_{i}y }[/math] for all [math]\displaystyle{ i = 1, 2, \ldots, t }[/math]. The dimension of a poset [math]\displaystyle{ P }[/math], denoted by [math]\displaystyle{ \dim P }[/math], is the smallest size of a realizer of [math]\displaystyle{ P }[/math].