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In the monoid (,) of the natural numbers with multiplication, only and are idempotent. Indeed, = and =. In the monoid (, +) of the natural numbers with addition, only is idempotent.
The idempotency of plays a role in other calculations as well, such as in determining the variance of the estimator ^. An idempotent linear operator P {\displaystyle P} is a projection operator on the range space R ( P ) {\displaystyle R(P)} along its null space N ( P ) {\displaystyle N(P)} .
For example, the relation < on the rational numbers is idempotent. The strict ordering relation is transitive, and whenever two rational numbers x and z obey the relation x < z there necessarily exists another rational number y between them (for instance, their average) with x < y and y < z.
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In ring theory, a branch of mathematics, an idempotent element or simply idempotent of a ring is an element a such that a 2 = a. [1] [a] That is, the element is idempotent under the ring's multiplication.
Idempotency of entailment is a property of logical systems that states that one may derive the same consequences from many instances of a hypothesis as from just one. This property can be captured by a structural rule called contraction, and in such systems one may say that entailment is idempotent if and only if contraction is an admissible rule.
and the principle of idempotency of conjunction: P ∧ P ⇔ P {\displaystyle P\land P\Leftrightarrow P} Where " ⇔ {\displaystyle \Leftrightarrow } " is a metalogical symbol representing "can be replaced in a logical proof with".
A square matrix is called a projection matrix if it is equal to its square, i.e. if =. [2]: p. 38 A square matrix is called an orthogonal projection matrix if = = for a real matrix, and respectively = = for a complex matrix, where denotes the transpose of and denotes the adjoint or Hermitian transpose of .