The lattice of semiretracts of a free monoid

The poset of retracts of a free monoid F is a lattice only when F is generated by three or fewer elements. We extend this result by widening attention from the retracts of F to the finite intersections of retracts, which we call semiretracts. When F is generated by three or fewer elements every semiretract is a retract. We obtain the desired generalization: The semiretracts of a finitely generated free monoid form a complete lattice. Moreover, each such lattice satisfies the ascending and descending chain conditions. These results are demonstrated through the use of special features of the minimal generating sets of retracts of free monoids.