Finite state wreath powers of transformation semigroups. (Q637615)

The author introduces the notion of a finite state wreath power \(FW^\infty(T,X)\) of a given transformation semigroup \((T,X)\) as a naturally defined subsemigroup in the projective limit of the finite wreath powers of \((T,X)\). The author establishes a criterion for semitransitivity of the wreath product of transformation semigroups and shows that \(FW^\infty(T,X)\) acts transitively on the set of ultimately periodic infinite words over \(X\) provided that \((T,X)\) is transitive. Further, the author shows that the finite state wreath power of a nontrivial transformation semigroup is not finitely generated. Finally, it is shown that the finite state wreath power of a finite transformation monoid containing a non-injective transformation does not contain irreducible generating systems.