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Let (A, ≤) and (B, ≤) be two partially ordered sets. A monotone Galois connection between these posets consists of two monotone [1] functions: F : A → B and G : B → A, such that for all a in A and b in B, we have F(a) ≤ b if and only if a ≤ G(b). In this situation, F is called the lower adjoint of G and G is called the upper adjoint ...
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
In mathematics, a relation denotes some kind of relationship between two objects in a set, which may or may not hold. [1] As an example, " is less than " is a relation on the set of natural numbers ; it holds, for instance, between the values 1 and 3 (denoted as 1 < 3 ), and likewise between 3 and 4 (denoted as 3 < 4 ), but not between the ...
Since sets are objects, the membership relation can relate sets as well, i.e., sets themselves can be members of other sets. A derived binary relation between two sets is the subset relation, also called set inclusion. If all the members of set A are also members of set B, then A is a subset of B, denoted A ⊆ B.
It associates a complete lattice to any binary relation between two sets by constructing a Galois connection from the relation, which then leads to two dually isomorphic closure systems. [5] Closure systems are intersection-closed families of sets. When ordered by the subset relation ⊆, they are complete lattices.
A pair of adjoint functors between two partially ordered sets is called a Galois connection (or, if it is contravariant, an antitone Galois connection). See that article for a number of examples: the case of Galois theory of course is a leading one.
Each ellipse is a connected set, but the union is not connected, since it can be partitioned to two disjoint open sets and . This means that, if the union X {\displaystyle X} is disconnected, then the collection { X i } {\displaystyle \{X_{i}\}} can be partitioned to two sub-collections, such that the unions of the sub-collections are disjoint ...
Given two sets and , the set of binary relations between them (,) can be equipped with a ternary operation [,,] = where denotes the converse relation of . In 1953 Viktor Wagner used properties of this ternary operation to define semiheaps , heaps, and generalized heaps.