All definitions tacitly require the homogeneous relation R {\displaystyle R} be transitive: for all a , b , c , {\displaystyle a,b,c,} if a R b {\displaystyle aRb} and b R c {\displaystyle bRc} then a R c . {\displaystyle aRc.} A term's definition may require additional properties that are not listed in this table.
In set theory, a prewellordering on a set X {\displaystyle X} is a preorder ≤ {\displaystyle \leq } on X {\displaystyle X} (a transitive and reflexive relation on X {\displaystyle X} ) that is strongly connected (meaning that any two points are comparable) and well-founded in the sense that the induced relation x < y {\displaystyle x<y} defined by x ≤ y and y ≰ x {\displaystyle x\leq y{\text{ and }}y\nleq x} is a well-founded relation.
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