Binary operation on magma determines neutral element

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Suppose (S,*) is a magma (set S with associative binary operation *). Then, if there exists a neutral element for * (i.e., an element e such that e * a = a * e = a for all a \in S), the element e 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.