Ordered set is used with distinct meanings in order theory

• A set with a binary relation R on its elements that is reflexive (for all a in the set aRa), antisymmetric (if aRb and bRa then a=b) and transitive (if aRb and bRc then aRc) is described as a partially ordered set or poset.
• If the binary relation is antisymmetric, transitive and also total (for all a and b in the set, aRb or bRa) then the set is a totally ordered set.
• If every non-empty subset has a least element then the set is a well-ordered set.