Concept

# Linear extension

Summary
In order theory, a branch of mathematics, a linear extension of a partial order is a total order (or linear 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 Linear extension of a partial order A partial order is a reflexive, transitive and antisymmetric relation. Given any partial orders ,\leq, and ,\leq^, on a set X, ,\leq^, is a linear extension of ,\leq, exactly when

# For every x, y \in X, if x \leq y, then x \leq^* y.

It is that second property that leads mathematicians to describe ,\leq^*, as extending ,\leq. Alternatively, a linear extension may be viewed as an order-preserving bijection from a partially ordered set P
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