In mathematics, and more specifically in order theory, several different types of ordered set have been studied.They include: * Cyclic orders, orderings in which triples of elements are either clockwise or counterclockwise * 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. * Partially ordered sets (or posets), orderings in which some pairs are comparable and others might not be * Preorders, a generalization of partial orders allowing ties (represented as equivalences and distinct from incomparabilities) * Semiorders, partial orders determined by comparison of numerical values, in which values that are too close to each other are incomparable; a