When we talk about an abstract binary operation *:SxS --> S We say that an identity element "e" exists when e * x = x * e = x, (x; e belong to S) Now my questions: 1) If the operation is not commutative, does not this imply no identity? since e * x != x * e necessarily? 2) Does e * x = x imply x * e = x? And if one side is true, can we still say that * has an identity restricted to one side? 3) Is there cases where not all x's of the set S have the same identity wrt operation *? These questions are regardless of any kind of algebraic structure. Thanks.