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

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.