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'').
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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.
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>
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<p>If ''P'' is a pre-function ''P'' : ''X'' ~> ''Y'' that happens to be total at ''X'', then ''P'' is known as a "function" from ''X'' to ''Y'', typically indicated as ''P'' : ''X'' → ''Y''.</p>
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<p>To say that a relation ''P'' ⊆ ''X'' × ''Y'' is totally tubular at ''X'' is to say that it is 1-regular at ''X''. Thus, we may formalize the following definitions:</p>
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To say that a relation ''P'' ⊆ ''X'' × ''Y'' is totally tubular at ''X'' is to say that it is 1-regular at ''X''. Thus, we may formalize the following definitions: