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If <math>L\!</math> is tubular at <math>X,\!</math> then <math>L\!</math> is called a ''partial function'' or a ''prefunction'' from <math>X\!</math> to <math>Y,\!</math> indicated by writing <math>L : X \rightharpoonup Y.\!</math>  We have the following definitions and notations.

If <math>L\!</math> is tubular at <math>X,\!</math> then <math>L\!</math> is known as a ''partial function'' or a ''prefunction'' from <math>X\!</math> to <math>Y,\!</math> indicated by writing <math>L : X \rightharpoonup Y.\!</math>  We have the following definitions and notations.

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<pre>

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> indicated by writing <math>L : X \to Y.\!</math>  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.

If R is a prefunction p : S ~> T that happens to be total at S, then R is called a "function" from S to T, indicated by writing R = f : S  > T.

 

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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.

R = f : S  > T iff R is 1 regular at S.

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f is surjective iff f is total at T.

f is injective iff f is tubular at T.

<math>\begin{array}{lll}

f is bijective iff f is 1 regular at T.

f ~\text{is surjective}

& \iff &

f ~\text{is total at}~ Y.

f ~\text{is injective}

& \iff &

f ~\text{is tubular at}~ Y.

f ~\text{is bijective}

& \iff &

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

A few more comments on terminology are needed in further preparation.  One of the constant practical demands encountered in this project is to have available a language and a calculus for relations that can permit discussion and calculation to range over functions, dyadic relations, and n place relations with a minimum amount of trouble in making transitions from subject to subject and in drawing the appropriate generalizations.

A few more comments on terminology are needed in further preparation.  One of the constant practical demands encountered in this project is to have available a language and a calculus for relations that can permit discussion and calculation to range over functions, dyadic relations, and n place relations with a minimum amount of trouble in making transitions from subject to subject and in drawing the appropriate generalizations.