Partial order

Correspondence of strict and non-strict partial order relations

Strict and non-strict partial orders on a set ${\displaystyle P}$ are closely related. A non-strict partial order ${\displaystyle \leq }$ may be converted to a strict partial order by removing all relationships of the form ${\displaystyle a\leq a;}$ that is, the strict partial order is the set ${\displaystyle <\;:=\ \leq \ \setminus \ \Delta _{P}}$ where ${\displaystyle \Delta _{P}:=\{(p,p):p\in P\}}$ is the identity relation on ${\displaystyle P\times P}$ and ${\displaystyle \;\setminus \;}$ denotes set subtraction. Conversely, a strict partial order < on ${\displaystyle P}$ may be converted to a non-strict partial order by adjoining all relationships of that form; that is, ${\displaystyle \leq \;:=\;\Delta _{P}\;\cup \;<\;}$ is a non-strict partial order. Thus, if ${\displaystyle \leq }$ is a non-strict partial order, then the corresponding strict partial order < is the irreflexive kernel given by $${\displaystyle a<b{\text{ if }}a\leq b{\text{ and }}a\neq b.}$$ Conversely, if < is a strict partial order, then the corresponding non-strict partial order ${\displaystyle \leq }$ is the reflexive closure given by: $${\displaystyle a\leq b{\text{ if }}a<b{\text{ or }}a=b.}$$