In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by Pop or Pd. This dual order Pop is defined to be the same set, but with the inverse order, i.e. x ≤ y holds in Pop if and only if y ≤ x holds in P. It is easy to see that this construction, which can be depicted by flipping the Hasse diagram for P upside down, will indeed yield a partially ordered set. In a broader sense, two partially ordered sets are also said to be duals if they are dually isomorphic, i.e. if one poset is order isomorphic to the dual of the other.
The importance of this simple definition stems from the fact that every definition and theorem of order theory can readily be transferred to the dual order. Formally, this is captured by the Duality Principle for ordered sets:
If a statement or definition is equivalent to its dual then it is said to be self-dual. Note that the consideration of dual orders is so fundamental that it often occurs implicitly when writing ≥ for the dual order of ≤ without giving any prior definition of this "new" symbol.
Naturally, there are a great number of examples for concepts that are dual:
Examples of notions which are self-dual include:
Since partial orders are antisymmetric, the only ones that are self-dual are the equivalence relations.
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra.
In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." The word partial in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable.
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). Specifically, every non-empty finite lattice is complete. Complete lattices appear in many applications in mathematics and computer science. Being a special instance of lattices, they are studied both in order theory and universal algebra.
In mathematics, a Heyting algebra is a bounded lattice equipped with a binary operation a → b of implication such that ≤ b is equivalent to c ≤. From a logical standpoint, A → B is by this definition the weakest proposition for which modus ponens, the inference rule A → B, A ⊢ B, is sound. Like Boolean algebras, Heyting algebras form a variety axiomatizable with finitely many equations. Heyting algebras were introduced by Arend Heyting (1930) to formalize intuitionistic logic.
In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection. Indeed, these lattices of sets describe the scenery completely: every distributive lattice is—up to isomorphism—given as such a lattice of sets.
In mathematics, the Boolean prime ideal theorem states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement for filters on sets is known as the ultrafilter lemma. Other theorems are obtained by considering different mathematical structures with appropriate notions of ideals, for example, rings and prime ideals, or distributive lattices and maximal ideals. This article focuses on prime ideal theorems from order theory.
Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary.
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:
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every two elements have a unique supremum and a unique infimum. An example is given by the natural numbers, partially ordered by divisibility, for which the unique supremum is the least common multiple and the unique infimum is the greatest common divisor.
In mathematical order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion of a ring ideal of abstract algebra, it has subsequently been generalized to a different notion. Ideals are of great importance for many constructions in order and lattice theory.
In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set (poset). The most familiar example is the completeness of the real numbers. A special use of the term refers to complete partial orders or complete lattices. However, many other interesting notions of completeness exist.
In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points, so that the dual of A is A itself. For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry.
In the mathematical area of order theory, there are various notions of the common concept of distributivity, applied to the formation of suprema and infima. Most of these apply to partially ordered sets that are at least lattices, but the concept can in fact reasonably be generalized to semilattices as well.
In the mathematical area of order theory, the compact elements or finite elements of a partially ordered set are those elements that cannot be subsumed by a supremum of any non-empty directed set that does not already contain members above the compact element. This notion of compactness simultaneously generalizes the notions of finite sets in set theory, compact sets in topology, and finitely generated modules in algebra.
In mathematics, a join-semilattice is a partially ordered set that has a join for any nonempty finite subset. Dually, a meet-semilattice is a partially ordered set which has a meet for any nonempty finite subset. Every join-semilattice is a meet-semilattice in the inverse order and vice versa.
In abstract algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are to topology and the modal logic S4 what Boolean algebras are to set theory and ordinary propositional logic. Interior algebras form a variety of modal algebras.
In mathematics, specifically order theory, the join of a subset S of a partially ordered set P is the supremum of S, denoted ⋁S, and similarly, the meet of S is the infimum, denoted ⋀S. In general, the join and meet of a subset of a partially ordered set need not exist. Join and meet are dual to one another with respect to order inversion.
Boolean algebra is a mathematically rich branch of abstract algebra. Just as group theory deals with groups, and linear algebra with vector spaces, Boolean algebras are models of the equational theory of the two values 0 and 1. Common to Boolean algebras, groups, and vector spaces is the notion of an algebraic structure, a set closed under zero or more operations satisfying certain equations.
In mathematics, a refinement monoid is a commutative monoid M such that for any elements a0, a1, b0, b1 of M such that a0+a1=b0+b1, there are elements c00, c01, c10, c11 of M such that a0=c00+c01, a1=c10+c11, b0=c00+c10, and b1=c01+c11.
In mathematics, specifically order theory, the Dedekind–MacNeille completion of a partially ordered set is the smallest complete lattice that contains it. It is named after Holbrook Mann MacNeille whose 1937 paper first defined and constructed it, and after Richard Dedekind because its construction generalizes the Dedekind cuts used by Dedekind to construct the real numbers from the rational numbers. It is also called the completion by cuts or normal completion.