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:
![]() |
(39) |
We observe that is a function at
and we record this fact in either of the manners
or
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