Among the variety of regularities affecting dyadic relations we pay special attention to the -regularity conditions where is equal to

Let be an arbitrary dyadic relation. The following properties of can then be defined:

We previously examined dyadic relations that separately exemplified each of these regularity conditions. And we introduced a few bits of terminology and special-purpose notations for working with tubular relations:

We arrive by way of this winding stair at the special stamps of dyadic relations that are variously described as -regular, total and tubular, or total prefunctions on specified domains, either or or both, and that are more often celebrated as functions on those domains.

If is a pre-function that happens to be total at then is known as a function from to typically indicated as

To say that a relation is total and tubular at is to say that is -regular at Thus, we may formalize the following definitions:

For example, let and let be the dyadic relation depicted in the bigraph below:

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We observe that is a function at and we record this fact in either of the manners or