Connected relation

In mathematics, a relation on a set is called connected or complete or total if it relates (or "compares") all distinct pairs of elements of the set in one direction or the other while it is called strongly connected if it relates all pairs of elements. As described in the terminology section below, the terminology for these properties is not uniform. This notion of "total" should not be confused with that of a total relation in the sense that for all ${\displaystyle x\in X}$ there is a ${\displaystyle y\in X}$ so that ${\displaystyle x\mathrel {R} y}$ (see serial relation).

Transitive binary relations
Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric Total, Semiconnex Anti- reflexive Equivalence relation Y ✗ ✗ ✗ ✗ ✗ Y ✗ ✗ Preorder (Quasiorder) ✗ ✗ ✗ ✗ ✗ ✗ Y ✗ ✗ Partial order ✗ Y ✗ ✗ ✗ ✗ Y ✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering ✗ Y Y Y ✗ ✗ Y ✗ ✗ Lattice ✗ Y ✗ ✗ Y Y Y ✗ ✗ Join-semilattice ✗ Y ✗ ✗ Y ✗ Y ✗ ✗ Meet-semilattice ✗ Y ✗ ✗ ✗ Y Y ✗ ✗ Strict partial order ✗ Y ✗ ✗ ✗ ✗ ✗ Y Y Strict weak order ✗ Y ✗ ✗ ✗ ✗ ✗ Y Y Strict total order ✗ Y Y ✗ ✗ ✗ ✗ Y Y Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric Definitions, for all ${\displaystyle a,b}$ and ${\displaystyle S\neq \varnothing$ :} ${\displaystyle {\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}}$ ${\displaystyle {\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}}$ ${\displaystyle {\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \begin{align}\min S \\ \text{exists}\end{align}} ${\displaystyle {\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}}$ ${\displaystyle {\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}}$ ${\displaystyle aRa}$ ${\displaystyle {\text{not }}aRa}$ ${\displaystyle {\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}}$
Y indicates that the column's property is always true the row's term (at the very left), while ✗ indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively. All definitions tacitly require the homogeneous relation ${\displaystyle R}$ be transitive: for all ${\displaystyle a,b,c,}$ if ${\displaystyle aRb}$ and ${\displaystyle bRc}$ then ${\displaystyle aRc.}$ A term's definition may require additional properties that are not listed in this table.

Connectedness features prominently in the definition of total orders: a total (or linear) order is a partial order in which any two elements are comparable; that is, the order relation is connected. Similarly, a strict partial order that is connected is a strict total order. A relation is a total order if and only if it is both a partial order and strongly connected. A relation is a strict total order if, and only if, it is a strict partial order and just connected. A strict total order can never be strongly connected (except on an empty domain).