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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.

We arrive by way of this winding stair at the special stamps of 2-adic relations <math>L \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.

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>

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

To say that a relation <math>P \subseteq X \times Y</math> is ''totally tubular'' at <math>X\!</math> is to say that <math>P\!</math> is 1-regular at <math>X.\!</math>  Thus, we may formalize the following definitions:

To say that a relation <math>L \subseteq X \times Y\!</math> is ''totally tubular'' at <math>X\!</math> is to say that <math>L\!</math> is 1-regular at <math>X.\!</math>  Thus, we may formalize the following definitions:

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<math>\begin{array}{lll}

<math>\begin{array}{lll}

P ~\text{is a function}~ P : X \to Y

L ~\text{is a function}~ L : X \to Y

& \iff &

& \iff &

P ~\text{is}~ 1\text{-regular at}~ X.

L ~\text{is}~ 1\text{-regular at}~ X.

\\[6pt]

\\[6pt]

P ~\text{is a function}~ P : X \leftarrow Y

L ~\text{is a function}~ L : X \leftarrow Y

& \iff &

& \iff &

P ~\text{is}~ 1\text{-regular at}~ Y.

L ~\text{is}~ 1\text{-regular at}~ Y.

\end{array}</math>

\end{array}</math>

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In the case of a 2-adic relation <math>F \subseteq X \times Y</math> that has the qualifications of a function <math>f : X \to Y,</math> there are a number of further differentia that arise:

In the case of a 2-adic relation <math>L \subseteq X \times Y</math> that has the qualifications of a function <math>f : X \to Y,</math> there are a number of further differentia that arise:

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{| align="center" cellspacing="6" width="90%"