List of order structures in mathematics

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In mathematics, and more particularly in order theory, several different types of ordered set have been studied. They include:

  • Total orders, orderings that specify, for every two distinct elements, which one is less than the other
  • Well-orders, total orders in which every non-empty subset has a least element
  • Weak orders, generalizations of total orders allowing ties (represented either as equivalences or, in strict weak orders, as transitive incomparabilities)
  • Partially ordered sets (or posets), orderings in which some pairs are comparable and others might not be
  • Semiorders, partial orders determined by comparison of numerical values, in which values that are too close to each other are incomparable; a subfamily of partial orders with certain restrictions
  • Interval orders, partial orders that can be realized by a set of intervals on the real line with order relation "is strictly to the right of"
  • Preorders, a generalization of partial orders allowing ties (represented as equivalences and distinct from incomparabilities)
  • Well-quasi-orderings, a class of preorders generalizing the well-orders
  • Lattices, partial orders in which each pair of elements has a greatest lower bound and a least upper bound. Many different types of lattice have been studied; see map of lattices for a list.
  • Cyclic orders, orderings in which triples of elements are either clockwise or counterclockwise

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