On images of subshifts under embeddings of symbolic varieties

Abstract

AbstractWe show that the image of a subshiftXunder various injective morphisms of symbolic algebraic varieties over monoid universes with algebraic variety alphabets is a subshift of finite type, respectively a sofic subshift, if and only if so isX. Similarly, letGbe a countable monoid and letA,Bbe Artinian modules over a ring. We prove that for every closed subshift submodule$\Sigma \subset A^G$and every injectiveG-equivariant uniformly continuous module homomorphism$\tau \colon \! \Sigma \to B^G$, a subshift$\Delta \subset \Sigma $is of finite type, respectively sofic, if and only if so is the image$\tau (\Delta )$. Generalizations for admissible group cellular automata over admissible Artinian group structure alphabets are also obtained.