Strict Order

A relation is a strict order on a set if it is

1. Irreflexive: does not hold for any .

2. Asymmetric: if , then does not hold.

3. Transitive: and implies .

Note that transitivity and irreflexivity combined imply that if holds, then does not.

A strict order is total if, for any , either , , or .

Every partial order induces a strict order

Similarly, every strict order induces a partial order