# Maximal free expression

(Difference between revisions)
 Revision as of 20:09, 5 October 2006 (edit) (Convert from HaWiki)← Previous diff Current revision (12:15, 18 May 2009) (edit) (undo)m Line 1: Line 1: - 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 $(S, \le)$ 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. 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. [[Category:Glossary]] [[Category:Glossary]]
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.