Editing Identity element (section)

==Properties==
In the example ''S'' = {''e,f''} with the equalities given, ''S'' is a [[semigroup]]. It demonstrates the possibility for {{math|(''S'', ∗)}} to have several left identities. In fact, every element can be a left identity. In a similar manner, there can be several right identities. But if there is both a right identity and a left identity, then they must be equal, resulting in a single two-sided identity. 

To see this, note that if {{mvar|l}} is a left identity and {{mvar|r}} is a right identity, then {{math|1=''l'' = ''l'' ∗ ''r'' = ''r''}}. In particular, there can never be more than one two-sided identity: if there were two, say {{mvar|e}} and {{mvar|f}}, then {{math|''e'' ∗ ''f''}} would have to be equal to both {{mvar|e}} and {{mvar|f}}.

It is also quite possible for {{math|(''S'', ∗)}} to have ''no'' identity element,<ref>{{harvtxt|McCoy|1973|p=22}}</ref> such as the case of even integers under the multiplication operation.<ref name=":0" /> Another common example is the [[cross product]] of [[Euclidean vector|vectors]], where the absence of an identity element is related to the fact that the [[Direction (geometry)|direction]] of any nonzero cross product is always [[orthogonal]] to any element multiplied. That is, it is not possible to obtain a non-zero vector in the same direction as the original. Yet another example of structure without identity element involves the additive [[semigroup]] of [[Positive number|positive]] [[natural number]]s.