Greatest fixed point

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In mathematics, more specifically in order theory, the greatest fixed point (gfp or GFP) of a function is the fixed point which is greater than or equal to all other fixed points, according to some partial order.

For example, the greatest fixed point of the real function

f(x) = x²

is x = 1 with the usual order on the real numbers. Fixed-point theorems can yield algorithms for locating the greatest fixed point, although least fixed points are more commonly used. Greatest and least fixed points may have desirable properties that arbitrary fixed points do not have.

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