product of the corresponding strict total orders). This is also a partial order. All three can similarly be defined for the Cartesian product of more than two sets. Applied to the vector space Rn, each of these make it an ordered vector space. See also examples of partially ordered sets. A real function of n real variables defined on a subset of Rn defines a strict weak order and a corresponding total preorder on that subset. Related structures A binary relation that is antisymmetric, transitive, and reflexive (but not necessarily total) is a partial order. A group with a compatible total order is a totally ordered group. There are only a few nontrivial structures that are (interdefinable as) reducts of a total order. Forgetting the orientation results in a betweenness relation. Forgetting the location of the ends results in a cyclic order. Forgetting both data results in a separation relation. See also – a downward total partial order Notes References George Grätzer (1971). Lattice theory: first concepts and distributive lattices. W. H. Freeman and Co. John G. Hocking and Gail S. Young (1961). Topology. Corrected reprint, Dover, 1988. External links Binary relations Order theory Set theory