Small monoids

We deal with all monoids having 2 elements. Let M = (A,*,e) be a monoid with two elements. Suppose that a is the unique element of M different from e. Then for a * a we have only two possibilities. Either a * a = e or a * a = a. This determines the multiplication * completely and we find two multiplication tables for M. They give rise to two distinct monoids, M1 and M2.

M1
*
e
a
e
e
a
a
a
e
     
M2
*
e
a
e
e
a
a
a
a

Both monoids can be realized on the set Z/2Z. Indeed, addition  (with e = 0) leads to M1, while  multiplication (with e = 1) leads to M2.