A neutral element is a special element of an algebraic structure . It is characterized by the fact that each element is mapped to itself through the link with the neutral element.
- left neutral , if is for everyone ,
- legally neutral , if is for everyone ,
- neutral if left neutral and right neutral.
If the connection is commutative , then the three terms match. But if it is not commutative, then there can be a right-neutral element that is not left-neutral, or a left-neutral element that is not right-neutral.
The symbol is often used for the link ; one then speaks of a multiplicatively written semigroup. A neutral element is then called a single element and is symbolized by. As is common with ordinary multiplication , the painting point can be left out in many situations .
A semigroup can also be noted additively by using the symbol for the link . A neutral element is then called a null element and is symbolized by.
- In the real numbers is ( zero ) the neutral element of addition and ( one ) the neutral element of multiplication , as and for any real number .
- In the ring of - matrices over a body , the zero matrix is the neutral element of the matrix addition and the identity matrix is the neutral element of the matrix multiplication .
- In a function space , the null function is the neutral element of addition and the one function is the neutral element of multiplication.
- In the case of vectors , the zero vector is the neutral element of the vector addition .
- In a formal language , the empty word is the neutral element of the concatenation of words.
- If a semigroup has both right-neutral and left-neutral elements, then all of these elements match and have exactly one neutral element. Because is and for everyone , then is .
- The neutral element of a monoid is clearly determined.
- But if a semigroup has no right-neutral element, then it can have several left-neutral elements. The simplest example is any set with at least two elements with the link . Each element is left neutral, but none is right neutral. Similarly, there are also semigroups with right-neutral, but without left-neutral elements.
- This can also occur when multiplying in rings. One example is the partial ring
- of the 2-by-2 matrices over any body . It is easy to calculate that is a non-commutative ring. Exactly the elements are left-neutral with regard to the multiplication
- with . According to what has been said above, the multiplication in cannot have any right-neutral elements.