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单词 Idempotency
释义

idempotency


If (S,*) is a magma, then an element xS is said to be idempotentMathworldPlanetmathPlanetmath if x*x=x.For example, every identity elementMathworldPlanetmath is idempotent, and in a group this is the only idempotent element.An idempotent element is often just called an idempotent.

If every element of the magma (S,*) is idempotent, then the binary operationMathworldPlanetmath * (or the magma itself) is said to be idempotent. For example, the and operationsMathworldPlanetmath in a latticeMathworldPlanetmath (http://planetmath.org/Lattice) are idempotent, because xx=x and xx=x for all x in the lattice.

A function f:DD is said to be idempotent if ff=f. (This is just a special case of the first definition above, the magma in question being (DD,), the monoid of all functions from D to D with the operation of function composition.) In other words, f is idempotent if and only if repeated application of f has the same effect as a single application: f(f(x))=f(x) for all xD. An idempotent linear transformation from a vector spaceMathworldPlanetmath to itself is called a projectionPlanetmathPlanetmath.

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更新时间:2025/5/4 1:15:32