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Thus, we arrive by way of this winding stair at the very special stamps of 2-adic relations ''P'' ⊆ ''X'' × ''Y'' that are "total prefunctions" at ''X'' (or ''Y''), "total and tubular" at ''X'' (or ''Y''), or "1-regular" at ''X'' (or ''Y''), more often celebrated as "functions" at ''X'' (or ''Y'').

We arrive by way of this winding stair at the special stamps of 2-adic relations <math>P \subseteq X \times Y</math> that are variously described as ''1-regular'', ''total and tubular'', or ''total prefunctions'' on specified domains, either <math>X\!</math> or <math>Y\!</math> or both, and that are more often celebrated as ''functions'' on those domains.

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If <math>P\!</math> is a pre-function <math>P : X \rightharpoonup Y</math> that happens to be total at <math>X,\!</math> then <math>P\!</math> is known as a ''function'' from <math>X\!</math> to <math>Y,\!</math>, typically indicated as <math>P : X \to Y.</math>

<p>If ''P'' is a pre-function ''P''&nbsp;:&nbsp;''X''&nbsp;~>&nbsp;''Y'' that happens to be total at ''X'', then ''P'' is known as a "function" from ''X'' to ''Y'', typically indicated as ''P''&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;''Y''.</p>

<p>To say that a relation ''P''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Y'' is totally tubular at ''X'' is to say that it is 1-regular at ''X''.  Thus, we may formalize the following definitions:</p>

To say that a relation ''P''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Y'' is totally tubular at ''X'' is to say that it is 1-regular at ''X''.  Thus, we may formalize the following definitions:

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| iff

| iff

| ''P'' is 1-regular at ''Y''.

| ''P'' is 1-regular at ''Y''.

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