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
uniquely determines a free idempotent monoid. 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 
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.