A free idempotent monoid on a set 
is the quotient monoid of the free monoid on 
by the smallest equivalence
relation compatible with multiplication and
imposing 
for every word 
. Equivalently, every map
from 
to an idempotent monoid
extends uniquely to a monoid homomorphism.
If the generating set of such a monoid
is finite, then so is the free idempotent monoid itself.
The number of elements in the monoid
depends on the size of the generating set, and the size of
the generating set determines a free idempotent monoid up
to isomorphism. On zero
letters, the free idempotent monoid has one element (the
identity element). With one letter, the free
idempotent monoid has two elements 
and 
. With two letters, it has seven
elements: 
, 
, 
, 
, 
, 
, and 
. In general, the numbers of
elements in the free idempotent monoids on 
, 1, ... letters are 1, 2, 7, 160, 332381, ... (OEIS
A005345). These are given by the analytic expression
 |
where 
is a binomial coefficient.
."
Math. Proc. Cambridge Philos. Soc. 48, 35-40, 1952.Lallement,
G.