Glossary of order theory

This is a glossary of some terms used in various branches of mathematics that are related to the fields of order, lattice, and domain theory. Note that there is a structured list of order topics available as well. Other helpful resources might be the following overview articles: completeness properties of partial orders distributivity laws of order theory preservation properties of functions between posets. In the following, partial orders will usually just be denoted by their carrier sets. As long as the intended meaning is clear from the context, will suffice to denote the corresponding relational symbol, even without prior introduction. Furthermore, < will denote the strict order induced by NOTOC Acyclic. A binary relation is acyclic if it contains no "cycles": equivalently, its transitive closure is antisymmetric. Adjoint. See Galois connection. Alexandrov topology. For a preordered set P, any upper set O is Alexandrov-open. Inversely, a topology is Alexandrov if any intersection of open sets is open. Algebraic poset. A poset is algebraic if it has a base of compact elements. Antichain. An antichain is a poset in which no two elements are comparable, i.e., there are no two distinct elements x and y such that x ≤ y. In other words, the order relation of an antichain is just the identity relation. Approximates relation. See way-below relation. Antisymmetric relation. A homogeneous relation R on a set X is antisymmetric, if x R y and y R x implies x = y, for all elements x, y in X. Antitone. An antitone function f between posets P and Q is a function for which, for all elements x, y of P, x ≤ y (in P) implies f(y) ≤ f(x) (in Q). Another name for this property is order-reversing. In analysis, in the presence of total orders, such functions are often called monotonically decreasing, but this is not a very convenient description when dealing with non-total orders. The dual notion is called monotone or order-preserving. Asymmetric relation. A homogeneous relation R on a set X is asymmetric, if x R y implies not y R x, for all elements x, y in X.