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Revision as of 20:02, 16 November 2012
, 20:02, 16 November 2012→6.31. Relations in General: markup
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+R is total at S iff R is ³1 regular at S.
−R is total at T iff R is ³1 regular at T.
−R is tubular at S iff R is £1 regular at S.
−R is tubular at T iff R is £1 regular at T.
−
|}
|}
−Returning to dyadic relations, it is useful to describe some familiar classes of objects in terms of their local and numerical incidence properties. Let <math>L \subseteq S \times T\!</math> be an arbitrary dyadic relation. The following properties of <math>L\!</math> can then be defined.
+Returning to dyadic relations, it is useful to describe some familiar classes of objects in terms of their local and numerical incidence properties. Let <math>L \subseteq X \times Y\!</math> be an arbitrary dyadic relation. The following properties of <math>L\!</math> can then be defined.
+{| align="center" cellspacing="8" width="90%"
+|
+<math>\begin{array}{lll}
+L ~\text{is total at}~ X
+& \iff &
+L ~\text{is}~ (\ge 1)\text{-regular}~ \text{at}~ X.
+\\[6pt]
+L ~\text{is total at}~ Y
+& \iff &
+L ~\text{is}~ (\ge 1)\text{-regular}~ \text{at}~ Y.
+\\[6pt]
+L ~\text{is tubular at}~ X
+& \iff &
+L ~\text{is}~ (\le 1)\text{-regular}~ \text{at}~ X.
+\\[6pt]
+L ~\text{is tubular at}~ Y
+& \iff &
+L ~\text{is}~ (\le 1)\text{-regular}~ \text{at}~ Y.
+\end{array}</math>
+|}
<pre>
<pre>
−If R is tubular at S, then R is called a "partial function" or "prefunction" from S to T, often indicated by writing R = p : S ~> T.
If R is tubular at S, then R is called a "partial function" or "prefunction" from S to T, often indicated by writing R = p : S ~> T.
