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A function that is not monotonic. In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. [1] [2] [3] This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.
If B is a poset, the set of functions A → B can be ordered by the pointwise order f ≤ g ↔ (∀x ∈ A) f(x) ≤ g(x). It can be shown that a monotone function f is residuated if and only if there exists a (necessarily unique) monotone function f +: B → A such that f o f + ≤ id B and f + o f ≥ id A, where id is the identity function.
A function that is absolutely monotonic on [,) can be extended to a function that is not only analytic on the real line but is even the restriction of an entire function to the real line. The big Bernshtein theorem : A function f ( x ) {\displaystyle f(x)} that is absolutely monotonic on ( − ∞ , 0 ] {\displaystyle (-\infty ,0]} can be ...
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 of F.
In order theory, a branch of mathematics, an order embedding is a special kind of monotone function, which provides a way to include one partially ordered set into another. Like Galois connections , order embeddings constitute a notion which is strictly weaker than the concept of an order isomorphism .
Given the algebras () and () of complex-valued continuous functions on compact Hausdorff spaces,, every positive map () is completely positive. The transposition of matrices is a standard example of a positive map that fails to be 2-positive.
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. [1] [2] [3] That is, a function : is open if for any open set in , the image is open in . Likewise, a closed map is a function that maps closed sets to closed sets.
For example, in theoretical computer science, least fixed points of monotonic functions are used to define program semantics, see Least fixed point § Denotational semantics for an example. Often a more specialized version of the theorem is used, where L is assumed to be the lattice of all subsets of a certain set ordered by subset inclusion .