# nLab total relation

## In higher category theory

A (binary) relation $\sim$ on a set $A$ is total if any two elements that are related in one order or the other:

$\forall \left(x,y:A\right),\phantom{\rule{thickmathspace}{0ex}}x\sim y\phantom{\rule{thickmathspace}{0ex}}\vee \phantom{\rule{thickmathspace}{0ex}}y\sim x$\forall (x, y: A),\; x \sim y \;\vee\; y \sim x

In the language of the $2$-poset-with-duals Rel of sets and relations, a relation $R:A\to A$ is total if its intersection with its reverse is the universal relation:

$A×A\subseteq R\cup {R}^{\mathrm{op}}$A \times A \subseteq R \cup R^{op}

Of course, this containment is in fact an equality.

A total relation is necessarily reflexive.

Note that an entire relation is sometimes called ‘total’, but these are unrelated concepts. The ‘total’ there is in the sense of a total (as opposed to partial) function, while the ‘total’ here is in the sense of total (as opposed to partial) order.

Revised on August 24, 2012 20:05:41 by Urs Schreiber (89.204.138.8)