# Binary operation on magma determines neutral element

From Groupprops

## Statement

Suppose is a magma (set with associative binary operation ). Then, if there exists a neutral element for (i.e., an element such that for all ), the element is uniquely determined by .

In other words, a magma can have at most one two-sided neutral element.

## Facts used

Equality of left and right neutral element

## Related facts

In the case that is associative, this says that the identity element (neutral element) of a monoid is completely determined by the binary operation. This yields the fact that monoids form a non-full subcategory of semigroups.