The syntactic monoid of the semigroup generated by a maximal prefix code

In this paper we investigate the semigroup structure of the syntactic monoid $\mathrm {Syn}(C^+)$ of $C^+$, the semigroup generated by a maximal prefix code $C$ for which $C^+$ is a single class of the syntactic congruence. In particular we prove that for such a prefix code $C$, either $\mathrm {Syn}(C^+)$ is a group or it is isomorphic to a special type of submonoid of $G\times \mathcal{T}(R)$ where $G$ is a group and $\mathcal {T}(R)$ is the full transformation semigroup on a set $R$ with more than one element. From this description we conclude that $\mathrm {Syn}(C^+)$ has a kernel $J$ which is a right group. We further investigate separately the case when $J$ is a right zero semigroup and the case when $J$ is a group.