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