Does the identity element of a monoid automatically commute with the other elements?

Question

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?

Answer 1

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.)

Answer 2

The semigroups in which the identity $xy = y$ holds is called a right-zero band in the semigroup literature. In such a semigroup, every element is idempotent ($x^2 = x$) and a left identity, but is an identity only if the semigroup has only one element.

To answer your question "Are they useful for something?", the answer is yes. For instance, they are useful to describe the fine structure of the minimal ideal of a semigroup. They also play a role in the wreath product decomposition theory of finite semigroups.