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Home > Glossary of order theory

• 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 ≤.

1 A

• 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 orderIn mathematics, a total order or linear order on a set X is a binary relation that is reflexive, antisymmetric, transitive, and total . This means that, if we denote the relation by ≤, the following statements hold for all a b and c in X : a ≤ a (res, 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.

• An asymmetric relationIn mathematics a relation is a generalization of arithmetic relations such as " " and "<" which occur in statements such as "5 < 6" or "2 + 2 4". See relation (mathematics), binary relation and relational algebra. A relational database stores data in rela R is a relation that is not symmetric.

• 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.

2 B

• Base. See continuous poset.

• A Boolean algebraIn mathematics and computer science, Boolean algebras or Boolean lattices are algebraic structures which "capture the essence" of the logical operations AND, OR and NOT as well as the corresponding set theoretic operations intersection, union and compleme is a distributive lattice with least element 0 and greatest element 1, in which every element x has a complement ¬x, such that x ^ ¬x = 0 and x v ¬x = 1.

• A boundedThe term bounded appears in different parts of mathematics where a notion of "size" can be given. The basic intuitive meaning common to all of them is that something is of finite size, and that this is the case if it is smaller than some other object that poset is one that has a least element 0 and a greatest element 1.

• A poset is bounded completeIn the mathematical field of order theory, a partially ordered set is bounded complete if all of its subsets which have some upper bound also have a least upper bound. Such a partial order can also be called consistently complete since any upper bound of if every of its subsets with some upper bound also has a least such upper bound. The dual notion is not common.

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