Editing Binary relation (section)

== Fringe of a relation ==
Given a relation <math>R</math>, its '''fringe''' is the sub-relation defined as
<math display="block">\operatorname{fringe}(R) = R \cap \overline{R \bar{R}^\textsf{T} R}.</math>

When <math>R</math> is a partial identity relation, difunctional, or a block diagonal relation, then <math>\operatorname{fringe}(R) = R</math>. Otherwise the <math>\operatorname{fringe}</math> operator selects a boundary sub-relation described in terms of its logical matrix: <math>\operatorname{fringe}(R)</math> is the side diagonal if <math>R</math> is an upper right triangular [[linear order]] or [[strict order]]. <math>\operatorname{fringe}(R)</math> is the block fringe if <math>R</math> is irreflexive (<math>R \subseteq \bar{I}</math>) or upper right block triangular. <math>\operatorname{fringe}(R)</math> is a sequence of boundary rectangles when <math>R</math> is of Ferrers type.

On the other hand, <math>\operatorname{fringe}(R) = \emptyset</math> when <math>R</math> is a [[dense order|dense]], linear, strict order.<ref name=GS11>[[Gunther Schmidt]] (2011) ''Relational Mathematics'', pages 211−15, [[Cambridge University Press]] {{ISBN|978-0-521-76268-7}}</ref>