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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
# ,\leq^*, is a total order, and

# 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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2001