Changes

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>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>

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>

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:

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:

{| cellpadding="4"

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

| ''P'' is a "function" ''p'' : ''X'' &rarr; ''Y''

|

| iff

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

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

& \iff &

| ''P'' is a "function" ''p'' : ''X'' &larr; ''Y''

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

| iff

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

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

& \iff &

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

|}

|}