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Hence, an antitone function f satisfies the property (), for all x and y in its domain. A constant function is both monotone and antitone; conversely, if f is both monotone and antitone, and if the domain of f is a lattice, then f must be constant. Monotone functions are central in order theory.
The implications of the two definitions of Galois connections are very similar, since an antitone Galois connection between A and B is just a monotone Galois connection between A and the order dual B op of B. All of the below statements on Galois connections can thus easily be converted into statements about antitone Galois connections.
The set complement on a powerset is an example of an antitone function. An important question is when two orders are "essentially equal", i.e. when they are the same up to renaming of elements. Order isomorphisms are functions that define such a renaming. An order-isomorphism is a monotone bijective function that has a
A function f between posets P and Q is an order-embedding if, for all elements x, y of P, x ≤ y (in P) is equivalent to f (x) ≤ f (y) (in Q). Order isomorphism. A mapping f: P → Q between two posets P and Q is called an order isomorphism, if it is bijective and both f and f−1 are monotone functions.
In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the good convergence behaviour of monotonic sequences, i.e. sequences that are non- increasing, or non- decreasing. In its simplest form, it says that a non-decreasing bounded -above sequence of real numbers converges to ...
Discontinuities of monotone functions. In the mathematical field of analysis, a well-known theorem describes the set of discontinuities of a monotone real-valued function of a real variable; all discontinuities of such a (monotone) function are necessarily jump discontinuities and there are at most countably many of them.
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Operator monotone function. In linear algebra, the operator monotone function is an important type of real-valued function, fully classified by Charles Löwner in 1934. [1] It is closely allied to the operator concave and operator concave functions, and is encountered in operator theory and in matrix theory, and led to the Löwner–Heinz ...