Pseudovarieties defining classes of sofic subshifts closed under taking shift equivalent subshifts

Abstract

For a pseudovariety V of ordered semigroups, let S ( V ) be the class of sofic subshifts whose syntactic semigroup lies in V . It is proved that if V contains Sl − then S ( V ∗ D ) is closed under taking shift equivalent subshifts, and conversely, if S ( V ) is closed under taking conjugate subshifts then V contains LSl − and S ( V ) = S ( V ∗ D ) . Almost finite type subshifts are characterized as the irreducible elements of S ( LInv ) , which gives a new proof that the class of almost finite type subshifts is closed under taking shift equivalent subshifts.