Binary operation on magma determines neutral element
From Groupprops
Statement
Suppose is a magma (set
with 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.