quality:corruption
Constructions

Every binary relation R {\displaystyle R} on a set S {\displaystyle S} can be extended to a preorder on S {\displaystyle S} by taking the transitive closure and reflexive closure, R + =. {\displaystyle R^{+=}.} The transitive closure indicates path connection in R: x R + y {\displaystyle R:xR^{+}y} if and only if there is an R {\displaystyle R} - path from x {\displaystyle x} to y. {\displaystyle y.}

Left residual preorder induced by a binary relation

Given a binary relation R, {\displaystyle R,} the complemented composition R ∖ R = R T ∘ R ¯ ¯ {\displaystyle R\backslash R={\overline {R^{\textsf {T}}\circ {\overline {R}}}}} forms a preorder called the left residual, where R T {\displaystyle R^{\textsf {T}}} denotes the converse relation of R, {\displaystyle R,} and R ¯ {\displaystyle {\overline {R}}} denotes the complement relation of R, {\displaystyle R,} while ∘ {\displaystyle \circ } denotes relation composition.