Maximal free expression

A free expression which is as large as it can be in the sense that is not a proper subexpression of another free express.

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 $(S,\leq )$ is called maximal if there is no $y\in S$ such that $x\leq y$, and it is called a maximum if $\forall y\in S,x\leq y$. If a maximum exists, it is unique, but there can be many maximal (but not maximum) elements.