On the Syntactic Monoids Associated with a Class of Synchronized Codes

Abstract

A complete codeCover an alphabetAis calledsynchronizedif there existx,y∈C*such thatxA*∩A*y⊆C*. In this paper we describe the syntactic monoidSyn(C+)ofC+for a complete synchronized codeCoverAsuch thatC+, the semigroup generated byC, is a single class of its syntactic congruencePC+. In particular, we prove that, for such a codeC, eitherC=AorSyn(C+)is isomorphic to a special submonoid of𝒯l(I)×𝒯r(Λ), where𝒯l(I)and𝒯r(Λ)are the full transformation semigroups on the nonempty setsIandΛ, respectively.