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Maximal free expression

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