entire relation

A binary relation from a set XX to a set YY is called entire if every element of XX is related to at least one element of YY. 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 called total, although that has another meaning in the theory of partial orders; see total relation.

A function is precisely a relation that is both entire and functional.

Like any relation, an entire relation rr can be viewed as a span

Γ r π r ϕ r X Y \array { & & \Gamma_r \\ & \swarrow_{\pi_r} & & \searrow^{\phi_r} \\ X & & & & Y }

Such a span is a relation iff the pairing map from the graph Γ r\Gamma_r to X×YX \times Y is an injection, and such a relation is entire iff the projection map π r\pi_r is a surjection.

The axiom of choice says precisely that every entire relation contains a function. Failing that, the COSHEP axiom may be interpreted to say that, given XX, there is a single surjection π X:Γ XX\pi_X: \Gamma_X \to X such that every entire relation from XX contains a relation given by a span whose left leg is π X\pi_X. In any case, entire relations may be preferable to functions in some contexts where the axiom of choice fails.

When internalising entire relations to a site, one may want to replace the projection map π r:Γ rX\pi_r: \Gamma_r \to X by a covering family.

Last revised on August 24, 2012 at 20:06:32. See the history of this page for a list of all contributions to it.