I was studying group theory and got into thinking about "groups without inverse elements", which are apparently called monoids. In every definition of monoid (and group) that I was able to find, the existence of the identity element is stated in this way:
Definition 1: There is $e \in G$ s.t. $e a = a e = a$ for all $a \in G$.
Now suppose we kept the other monoid axioms and changed this statement to:
Definition 2: There is $e \in G$ s.t. $e a = a$ for all $a \in G$.
Are definitions 1 and 2 equivalent (when taken together with the other two monoid axioms)? I wasn't able to show that they are (if an inverse of $a$ exists, i.e. we are talking about groups, then they clearly are). If not, is there a name for the construction that comes out using definition 2? Some kind of monoid with a non-commuting identity element? Are they useful for something?
Best Answer
Consider a set $S$ with $|S|>1$ bestowed with the binary operation $\circ$ where:
$$\forall x, y\in S: x \circ y=y$$
Then every element of $S$ is a left identity, and there are no right identities.
(Here $(S,\circ)$ is a semigroup but not a monoid, as associativity holds.)