A geometric description of the maximal monoids of some matrix semigroups

The maximal monoids of the form FSF are studied, where F is a nonnegative idempotent matrix and S is one of the following matrix semigroups: Nn, the nonnegative square matrices, Stn, the stochastic matrices, and Dn, the doubly stochastic matrices. For the cases of Nn and Stn, it is shown that these maximal monoids are affinely isomorphic to the full semigroups of lower order, and for FDnF that it is a compact affine semigroup with zero, here called the core of a primitive idempotent.