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the identity function is idempotent; the floor, ceiling and fractional part functions are idempotent; the real part function () of a complex number, is idempotent. the subgroup generated function from the power set of a group to itself is idempotent; the convex hull function from the power set of an affine space over the reals to itself is ...
In mathematics, an idempotent binary relation is a binary relation R on a set X (a subset of Cartesian product X × X) for which the composition of relations R ∘ R is the same as R. [ 1 ] [ 2 ] This notion generalizes that of an idempotent function to relations.
An idempotent a + I in the quotient ring R / I is said to lift modulo I if there is an idempotent b in R such that b + I = a + I. An idempotent a of R is called a full idempotent if RaR = R. A separability idempotent; see Separable algebra. Any non-trivial idempotent a is a zero divisor (because ab = 0 with neither a nor b being zero, where b ...
Idempotent matrices arise frequently in regression analysis and econometrics. For example, in ordinary least squares, the regression problem is to choose a vector β of coefficient estimates so as to minimize the sum of squared residuals (mispredictions) e i: in matrix form,
In mathematics, a band (also called idempotent semigroup) is a semigroup in which every element is idempotent (in other words equal to its own square). Bands were first studied and named by A. H. Clifford ( 1954 ).
One example of a Boolean ring is the power set of any set X, where the addition in the ring is symmetric difference, and the multiplication is intersection. As another example, we can also consider the set of all finite or cofinite subsets of X , again with symmetric difference and intersection as operations.
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.
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