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==Relations==
Next let's re-examine the ''numerical incidence properties'' of relations, concentrating on the definitions of the assorted regularity conditions.
For example, <math>L\!</math> is said to be <math>^{\backprime\backprime} c\text{-regular at}~ j \, ^{\prime\prime}</math> if and only if the cardinality of the local flag <math>L_{x \,\text{at}\, j}</math> is equal to <math>c\!</math> for all <math>x \in X_j,</math> coded in symbols, if and only if <math>|L_{x \,\text{at}\, j}| = c</math> for all <math>x \in X_j.</math>
In a similar fashion, it is possible to define the numerical incidence properties <math>^{\backprime\backprime}(< c)\text{-regular at}~ j \, ^{\prime\prime},</math> <math>^{\backprime\backprime}(> c)\text{-regular at}~ j \, ^{\prime\prime},</math> and so on. For ease of reference, a few of these definitions are recorded below.
{| align="center" cellspacing="6" width="90%"
|
<math>\begin{array}{lll}
L ~\text{is}~ c\text{-regular at}~ j
& \iff &
|L_{x \,\text{at}\, j}| = c ~\text{for all}~ x \in X_j.
\\[6pt]
L ~\text{is}~ (< c)\text{-regular at}~ j
& \iff &
|L_{x \,\text{at}\, j}| < c ~\text{for all}~ x \in X_j.
\\[6pt]
L ~\text{is}~ (> c)\text{-regular at}~ j
& \iff &
|L_{x \,\text{at}\, j}| > c ~\text{for all}~ x \in X_j.
\\[6pt]
L ~\text{is}~ (\le c)\text{-regular at}~ j
& \iff &
|L_{x \,\text{at}\, j}| \le c ~\text{for all}~ x \in X_j.
\\[6pt]
L ~\text{is}~ (\ge c)\text{-regular at}~ j
& \iff &
|L_{x \,\text{at}\, j}| \ge c ~\text{for all}~ x \in X_j.
\end{array}</math>
|}
Clearly, if any relation is <math>(\le c)\text{-regular}</math> on one of its domains <math>X_j\!</math> and also <math>(\ge c)\text{-regular}</math> on the same domain, then it must be <math>(= c)\text{-regular}\!</math> on that domain, in effect, <math>c\text{-regular}\!</math> at <math>j.\!</math>
Among the variety of conceivable regularities affecting 2-adic relations, we pay special attention to the <math>c\!</math>-regularity conditions where <math>c\!</math> is equal to 1.
Let <math>P \subseteq X \times Y</math> be an arbitrary 2-adic relation. The following properties of <math>~P~</math> can be defined:
{| align="center" cellspacing="6" width="90%"
|
<math>\begin{array}{lll}
P ~\text{is total at}~ X
& \iff &
P ~\text{is}~ (\ge 1)\text{-regular}~ \text{at}~ X.
\\[6pt]
P ~\text{is total at}~ Y
& \iff &
P ~\text{is}~ (\ge 1)\text{-regular}~ \text{at}~ Y.
\\[6pt]
P ~\text{is tubular at}~ X
& \iff &
P ~\text{is}~ (\le 1)\text{-regular}~ \text{at}~ X.
\\[6pt]
P ~\text{is tubular at}~ Y
& \iff &
P ~\text{is}~ (\le 1)\text{-regular}~ \text{at}~ Y.
\end{array}</math>
|}
We have already looked at 2-adic relations that separately exemplify each of these regularities. We also introduced a few bits of additional terminology and special-purpose notations for working with tubular relations:
{| align="center" cellspacing="6" width="90%"
|
<math>\begin{array}{lll}
P ~\text{is a pre-function}~ P : X \rightharpoonup Y
& \iff &
P ~\text{is tubular at}~ X.
\\[6pt]
P ~\text{is a pre-function}~ P : X \leftharpoonup Y
& \iff &
P ~\text{is tubular at}~ Y.
\end{array}</math>
|}
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>
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:
{| align="center" cellspacing="6" width="90%"
|
<math>\begin{array}{lll}
P ~\text{is a function}~ P : X \to Y
& \iff &
P ~\text{is}~ 1\text{-regular at}~ X.
\\[6pt]
P ~\text{is a function}~ P : X \leftarrow Y
& \iff &
P ~\text{is}~ 1\text{-regular at}~ Y.
\end{array}</math>
|}
==Table Work==
==Table Work==
