The syntactic monoid of an infix code

Abstract:

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