Editing Inverse element (section)

===Identity elements===
Let <math>*</math> be a possibly [[partial operation|partial]] associative operation on a set {{mvar|X}}.

An ''[[identity element]]'', or simply an ''identity'' is an element {{mvar|e}} such that 
:<math>x*e=x \quad\text{and}\quad e*y=y</math>
for every {{mvar|x}} and {{mvar|y}} for which the left-hand sides of the equalities are defined.

If {{mvar|e}} and {{mvar|f}} are two identity elements such that <math>e*f</math> is defined, then <math>e=f.</math> (This results immediately from the definition, by <math>e=e*f=f.</math>)

It follows that a total operation has at most one identity element, and if {{mvar|e}} and {{mvar|f}} are different identities, then <math>e*f</math> is not defined. 

For example, in the case of [[matrix multiplication]], there is one {{math|''n''×''n''}} [[identity matrix]] for every positive integer {{mvar|n}}, and two identity matrices of different size cannot be multiplied together.

Similarly, [[identity function]]s are identity elements for [[function composition]], and the composition of the identity functions of two different sets are not defined.