A binaryrelation from aset$X$ to a set$Y$ is calledentire if every element of$X$ is related to at least one element of$Y$. This includes most examples of what the pre-Bourbaki literature calls a (total)multi-valued function (although that term usually implied some continuity or analyticity properties as well). An entire relation is sometimes calledtotal, although that has another meaning in the theory ofpartial orders; seetotal relation.

Afunction is precisely a relation that is both entire andfunctional.

Like any relation, an entire relation$r$ can be viewed as aspan

Such a span is a relation iff the pairing map from thegraph${\Gamma }_r$ to$X\times Y$ is aninjection, and such a relation is entire iff theprojection map${\pi }_r$ is asurjection.

Theaxiom of choice says precisely that every entire relation contains afunction. Failing that, theCOSHEP axiom may be interpreted to say that, given$X$, there is a single surjection${\pi }_X:{\Gamma }_X\to X$ such that every entire relation from$X$ contains a relation given by a span whose left leg is${\pi }_X$. In any case, entire relations may be preferable to functions in some contexts where the axiom of choice fails.

Wheninternalising entire relations to asite, one may want to replace the projection map${\pi }_r:{\Gamma }_r\to X$ by acovering family.

Related concepts

• axiom of fullness

Last revised on October 15, 2023 at 18:34:24. See thehistory of this page for a list of all contributions to it.