# 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 express. |
+ | 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 <math>(S, \le)</math> is called '''maximal''' if there is no <math>y \in S</math> such that <math>x \le y</math>, and it is called a '''maximum''' if <math>\forall y \in S, x \le y</math>. If a maximum exists, it is unique, but there can be many maximal (but not maximum) elements. |
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 <math>(S, \le)</math> is called '''maximal''' if there is no <math>y \in S</math> such that <math>x \le y</math>, and it is called a '''maximum''' if <math>\forall y \in S, x \le y</math>. If a maximum exists, it is unique, but there can be many maximal (but not maximum) elements. |

## Latest revision as of 12:15, 18 May 2009

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 is called **maximal** if there is no such that , and it is called a **maximum** if . If a maximum exists, it is unique, but there can be many maximal (but not maximum) elements.