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In this paper we investigate the semigroup structure of the syntactic monoid 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 Syn(C+) is a group or it is isomorphic to a special type of submonoid of G× T (R) where G is a group and T (R) is the full transformation semigroup on a set R with more than one element. From this description we conclude that 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.

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