Neutral element

The neutral element or identity of a set Aendowed with an internal binary operation          {displaystyle circledast } :

:A× × AΔ Δ A(a,b) c=a b{displaystyle {begin{array}{rccl}circledast: strangerAtimes A fakelongrightarrow &A age-old dream(a,b) strangerlongmapsto &c=acircledast bend{array}}}}}}}

is an element e of the whole Afor any other element a of A, it is fulfilled:

a e=e a=a{displaystyle acircledast e=ecircledast a=a}

That is, a neutral element has a neutral effect when used in the operation {displaystyle circledast }. When operating any element of the set with the neutral element the result is the original element.

An element e to fulfill only e a=a{displaystyle ecircledast a=a} It's called neutral element on the left. Similarly an element f that fulfills only a f=a{displaystyle acircledast f=a} is called or called neutral element on the right. Such elements must not be equal, except for a group. There can be both, one of them or none in the case of a set provided with an operation.

Examples