If is a magma, then an element is said to be idempotent if . For example, every identity element 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 is idempotent, then the binary operation (or the magma itself) is said to be idempotent. For example, the and operations in a lattice (http://planetmath.org/Lattice) are idempotent, because and for all in the lattice.
A function is said to be idempotent if . (This is just a special case of the first definition above, the magma in question being , the monoid of all functions from to with the operation of function composition.) In other words, is idempotent if and only if repeated application of has the same effect as a single application: for all . An idempotent linear transformation from a vector space to itself is called a projection.
|Date of creation||2013-03-22 12:27:31|
|Last modified on||2013-03-22 12:27:31|
|Last modified by||yark (2760)|