## Relation Theory • 6

### Relation Theory • Species of Dyadic Relations

Returning to 2-adic relations, it is useful to describe several familiar classes of objects in terms of their local and numerical incidence properties.  Let $L \subseteq S \times T$ be an arbitrary 2-adic relation.  The following properties of $L$ can be defined.

If $L \subseteq S \times T$ is tubular at $S$ then $L$ is called a partial function or a prefunction from $S$ to $T.$  This is sometimes indicated by giving $L$ an alternate name, for example, ${}^{\backprime\backprime} p {}^{\prime\prime},$ and writing $L = p : S \rightharpoonup T.$  Thus we have the following definition.

$\begin{matrix} L & = & p : S \rightharpoonup T & \text{if and only if} & L & \text{is} & \text{tubular} & \text{at}~ S. \end{matrix}$

If $L$ is a prefunction $p : S \rightharpoonup T$ which happens to be total at $S,$ then $L$ is called a function from $S$ to $T,$ indicated by writing $L = f : S \to T.$  To say a relation $L \subseteq S \times T$ is totally tubular at $S$ is to say it is $1$-regular at $S.$  Thus, we may formalize the following definition.

$\begin{matrix} L & = & f : S \to T & \text{if and only if} & L & \text{is} & 1\text{-regular} & \text{at}~ S. \end{matrix}$

In the case of a function $f : S \to T,$ we have the following additional definitions.

### Resources

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