On the Syntactic Monoids Associated with a Class of Synchronized Codes

A complete code C over an alphabet A is called synchronized if there exist x, y ∈ C* such that xA*∩A*y⊆C*. In this paper we describe the syntactic monoid Syn(C (+)) of C (+) for a complete synchronized code C over A such that C (+), the semigroup generated by C, is a single class of its syntactic congruence P (C(+)). In particular, we prove that, for such a code C, either C = A or Syn(C (+)) is isomorphic to a special submonoid of 𝒯 (l)(I) × 𝒯 (r)(Λ), where 𝒯 (l)(I) and 𝒯 (r)(Λ) are the full transformation semigroups on the nonempty sets I and Λ, respectively.