Necessary and sufficient conditions on a monoid M M are found in order that M M be isomorphic to the syntactic monoid of a language L L over an alphabet X X having one of the following properties. In the first theorem L L is a P L {P_L} -class and P W ( L ) ⊆ P L {P_{W\left ( L \right )}} \subseteq {P_L} where P L {P_L} is the syntactic congruence of L L and W ( L ) W\left ( L \right ) is the residue of L L . In the second theorem L L is an infix code; that is, satisfies u , u v w ∈ L u,uvw \in L implying u = w = 1 u = w = 1 . In the third theorem L L is an infix code satisfying a condition which amounts to the requirement that M M be a nilmonoid. Various refinements of these conditions are also considered.