Glossary of Order Theory - A

A

• 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.

• A 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.

• 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. A relation R on a set X is asymmetric, if x R y implies not y R x, for all elements x, y in X.

• An atom in a poset P with least element 0, is an element that is minimal among all elements that are unequal to 0.

• A atomic poset P with least element 0 is one in which, for every non-zero element x of P, there is an atom a of P with a ≤ x.