Irreflexive kernel

Definitions

The reflexive reduction or irreflexive kernel of ${\displaystyle R}$ is the smallest (with respect to ${\displaystyle \subseteq }$) relation on ${\displaystyle X}$ that has the same reflexive closure as ${\displaystyle R.}$ It is equal to ${\displaystyle R\setminus \operatorname {I} _{X}=\{(x,y)\in R~:~x\neq y\}.}$ The reflexive reduction of ${\displaystyle R}$ can, in a sense, be seen as a construction that is the "opposite" of the reflexive closure of ${\displaystyle R.}$ For example, the reflexive closure of the canonical strict inequality ${\displaystyle <}$ on the reals${\displaystyle \mathbb {R} }$ is the usual non-strict inequality ${\displaystyle \leq }$ whereas the reflexive reduction of ${\displaystyle \leq }$ is ${\displaystyle <.}$