Idempotent

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idempotent element

An element of a ring, semi-group or groupoid equal to its own square: . An idempotent is said to contain an idempotent (denoted by ) if . For associative rings and semi-groups, the relation is a partial order on the set of idempotent elements, called the natural partial order on . Two idempotents and of a ring are said to be orthogonal if . With every idempotent of a ring (and also with every system of orthogonal idempotents) there is associated the so-called Peirce decomposition of the ring. For an -ary algebraic relation , an element is said to be an idempotent if , where occurs times between the brackets.