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===Commentary Note 11.7===

===Commentary Note 11.7===

We come now to the very special cases of 2-adic relations that are known as functions.  It will serve a dual purpose on behalf of the exposition if we take the class of functions as a source of object examples to clarify the more abstruse concepts in the RIG material.

We come now to the very special cases of 2-adic relations that are known as ''functions''.  It will serve a dual purpose on behalf of the present exposition if we take the class of functions as a source of object examples to clarify the more abstruse concepts in the RIG material.

To begin, let's recall the definition of a ''local flag'':

To begin, let's recall the definition of a ''local flag'':

: ''L''_''x''.''j'' = { (''x''₁, …, ''x''_''j'', …, ''x''_''k'') ∈ ''L'' : ''x''_''j'' = ''x'' }.

| <math>L_{x \star j} = \{ (x_1, \ldots, x_j, \ldots, x_k) \in L : x_j = x \}.</math>

|}

In the case of a 2-adic relation ''L''&nbsp;&sube;&nbsp;''X''₁&nbsp;&times;&nbsp;''X''₂ = ''X''&nbsp;&times;&nbsp;''Y'', we can reap the benefits of a radical simplification in the definitions of the local flags.  Also in this case, we tend to denote ''L''_''u''.1 by "''L''_''u''.''X''" and ''L''_''v''.2 by "''L''_''v''.''Y''".

In the case of a 2-adic relation ''L''&nbsp;&sube;&nbsp;''X''₁&nbsp;&times;&nbsp;''X''₂ = ''X''&nbsp;&times;&nbsp;''Y'', we can reap the benefits of a radical simplification in the definitions of the local flags.  Also in this case, we tend to denote ''L''_''u''.1 by "''L''_''u''.''X''" and ''L''_''v''.2 by "''L''_''v''.''Y''".