Irreflexive version
A set X is partially ordered by R if:
 For elements x of X, we do not have xRx (R is irreflexive)
 We cannot have both xRy and yRx (R is antisymmetric)
 xRy and yRz implies xRz (R is transitive)
This is intuitively like "strictly less than"  $<$


Reflexive version
A set X is partially ordered by R if:
 For elements x of X, we do have xRx (R is reflexive)
 If xRy and yRx then x=y (R is antisymmetric)
 xRy and yRz implies xRz (R is transitive)
This is intuitively like "less than or equal to"  $\le$
