Languages Defined by Generalized Equality Sets

We consider the generalized equality sets which are of the form E G (a,g 1,g 2) = { w | g 1(w) = ag 2(w)}, determined by instances of the generalized Post Correspondence Problem, where the morphisms g 1 and g 2 are nonerasing and a is a letter. We are interested in the family consisting of the languages h(E G (J)), where h is a coding and J is a shifted equality set of the above form. We prove several closure properties for this family.