# Maximal free expression

This is within the context of a given expression, and subexpressions are partially ordered with respect to containment, and have finite length, so there will always be maximal (but possibly not unique) free (sub-)expressions. Note that there is a subtle but important difference between the words maximal and maximum. An element x of a partially ordered set $(S, \le)$ is called maximal if there is no $y \in S$ such that $x \le y$, and it is called a maximum if $\forall y \in S, x \le y$. If a maximum exists, it is unique, but there can be many maximal (but not maximum) elements.