TITLE: Non-associative, non-commutative binary operation with a identity
QUESTION [10 upvotes]: Can you give me few examples of binary operation that it is not associative, not commutative but has an identity element?

REPLY [3 votes]: Here's an example of an abelian group without associativity, inspired by an answer to this question. Consider the game of rock-paper-scissors: $R$ is rock, $P$ is paper, $S$ is scissors, and $1$ is fold/draw/indeterminate. Let $\ast$ be the binary operation "play". 
\begin{array}{r|cccc}
\ast & 1 & R & P & S\\
\hline
   1& 1 & R & P & S \\
   R & R& 1 & P & R\\
   P & P& P & 1 & S\\
   S & S& R & S & 1 .
\end{array}
The mutliplication table above defines a set of elements, with a binary operation, that is commutative and non-associative. Also, each element has an inverse (itself), and the identity exists.