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'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 ''a''R''a''), antisymmetric (if ''a''R''b'' and ''b''R''a'' then ''a''=''b'') and transitive (if ''a''R''b'' and ''b''R''c'' then ''a''R''c'') 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, ''a''R''b'' or ''b''R''a'') 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'.