Closure operators

thumb|On/Off buttons of a train's destination sign control panel. Pressing the On button (green) is an idempotent operation, since it has the same effect whether done once or multiple times. Likewise, pressing Off is idempotent. Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of places in abstract algebra (in particular, in the theory of projectors and closure operators) and functional programming (in whi

the smallest superset of a given set that is closed under a given operation

notion in topological vector spaces

given a subset S of a topological space X, the biggest set of points in S not part of the boundary of S

in a topological space, the smallest closed set containing a given set

In combinatorics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid axiomatically, the most significant being in terms of: independent sets; bases or circuits; rank functions; closure operators; and closed sets or flats. In the language of partially ordered sets, a finite simple matroid is equivalent to a geometric lattice.

concept in algebra

one of several theorems stating that, under certain conditions, a function f will have an argument x for which f(x) = x

operation on binary relations

pair of adjoint functors between two preordered sets seen as categories

operation on binary relations

partially ordered set in which all subsets have both a supremum and infimum

smallest affine subspace that contains a subset

mathematical operator

operation on binary relations

topology in which the intersection of any family of open sets is open

closure of the set of conjugate elements under the group operation