Neutral element

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.

definition

Be a magma (a quantity with a two-digit link ). Then an element is called ${\displaystyle (S,*)}$ ${\displaystyle e\in S}$

• left neutral , if is for everyone , ${\displaystyle e*a=a}$ ${\displaystyle a\in S}$
• legally neutral , if is for everyone , ${\displaystyle a*e=a}$ ${\displaystyle a\in S}$
• neutral if left neutral and right neutral. ${\displaystyle e}$

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.

A semigroup with a neutral element is called a monoid . If every element in also has an inverse element in , then is a group . ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle S}$

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 . ${\displaystyle *}$ ${\displaystyle \cdot }$ ${\displaystyle 1}$ ${\displaystyle \cdot }$

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. ${\displaystyle *}$ ${\displaystyle +}$ ${\displaystyle 0}$

Examples

• In the real numbers is ( zero ) the neutral element of addition and ( one ) the neutral element of multiplication , as and for any real number . ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 0+x=x+0=x}$ ${\displaystyle 1\cdot x=x\cdot 1=x}$ ${\displaystyle x}$
• 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 . ${\displaystyle n\times n}$
• 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.

properties

• 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 . ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle a*e=a}$ ${\displaystyle f*a=a}$ ${\displaystyle a\in S}$ ${\displaystyle f=f*e=e}$

• 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. ${\displaystyle a*b:=b}$

• This can also occur when multiplying in rings. One example is the partial ring

${\displaystyle R=\left\{\left.{\begin{pmatrix}a&b\\0&0\end{pmatrix}}\right|a,b\in K\right\}}$

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 ${\displaystyle K}$ ${\displaystyle R}$

${\displaystyle {\begin{pmatrix}1&x\\0&0\end{pmatrix}}}$

with . According to what has been said above, the multiplication in cannot have any right-neutral elements. ${\displaystyle x\in K}$ ${\displaystyle R}$

See also

• Absorbent element
• Unit (math)
• Identical figure
• Inverse element

Individual evidence

1. ^ Siegfried Bosch : Algebra. 7th, revised edition. Springer, Berlin et al. 2009, ISBN 978-3-540-92811-9 , p. 2.