On the Syntactic Monoids Associated with a Class of Synchronized Codes

Abstract

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.