A free idempotent monoid is a monoid that satisfies the identity x^2 = x and is generated by a set of elements. 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 the generating set uniquely determines a free idempotent monoid. On zero letters, the free idempotent monoid has one element (the identity). With one letter, the free idempotent monoid has two elements (1, a). With two letters, it has seven elements: (1, a, b, a b, b a, a b a, b a b). In general, the numbers of elements in the free idempotent monoids on n letters are 1, 2, 7, 160, 332381, ... (OEIS A005345). These are given by the analytic expression

monoid