Identity element

A neutral element is a special element of an algebraic structure. It is characterized in that each element is represented by the link with the neutral element to itself.

Definition

Be a magma ( a lot with a two-digit shortcut). Then is called an element

  • Left neutral if for all,
  • Fairly neutral if for all,
  • Neutral, if is left neutral and fairly neutral.

If the link is commutative, then match the three terms. If they do not, however, is commutative, then there can be a fairly neutral element, that is not left neutral, or a left neutral element, that is not quite neutral.

A semigroup with neutral element is called monoid. Has, in addition, each element in an inverse element in such a group.

Frequently for the shortcut icon is used, then one speaks of a multiplicative semigroup written. A neutral element is then called unit element and is symbolized by. As is usual also in the ordinary multiplication, the Malpunkt can be omitted in many situations.

A semigroup can also be additively note by the icon used for the shortcut. A neutral element is then called zero element and is symbolized by.

Examples

  • In the real numbers (zero) is the neutral element of addition and (one) is the neutral element of multiplication, because and for every real number.
  • Is a ring, it is a commutative group.
  • In the ring of matrices over a field is the zero matrix is the neutral element of matrix addition and the identity matrix is the neutral element of the matrix multiplication.
  • In a function space, the zero function is the neutral element of addition and one function is the neutral element of multiplication.
  • For vectors of the zero vector is the neutral element of vector addition.

Properties

  • If a semigroup has both right and left neutral neutral elements, then all agree these elements and has exactly one identity element. Because is and for all, then.
  • The neutral element of a monoid is therefore uniquely determined.
  • Has a semigroup but not a right neutral element, then they can have several left neutral. The simplest example is an arbitrary least two-element set with the link. It is each element left neutral, but none quite neutral. Similarly, there are also semigroups with fairly neutral, but left without neutral elements.
  • This can also occur in the multiplication in rings. An example is the part of the ring
Constant function Null vector Unit (ring theory) Absorbing element
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