Idempotents in Symmetric Semigroups

Abstract We count the number of idempotent elements in a certain section of the s symmetric semigroup S n on n letters. As a corollary of our result we have that every maximal principal right ideal of S n contains ∑ i=1 n−1 i n−i−1 ( n−2 i−1 )+( n−1 i ) idempotent elements. Let T r (1 ⩽ r ⩽ n − 1) be the set of all elements of S n of rank less than or equal to r , and let D r denote the set of all elements of S n of rank r . Then T r is a semigroup generated by the idempotent elements of D r . We shall obtain a maximal mutant of T n−1 = S n D n .