Upper set monoids and length preserving morphisms

Length preserving morphisms and inverse of substitutions are two well-studied operations on regular languages. Their connection with varieties generated by power monoids was established independently by Reutenauer and Straubing in 1979. More recently, an ordered version of this theory was proposed by Polák and by the authors. In this paper, we present an improved version of these results and obtain the following consequences. Given a variety of finite ordered monoids , let be the variety of finite ordered monoids generated by the upper set monoids of members of . Then . This contrasts with the known results for the unordered case: the operator corresponding to power monoids satisfies , but the varieties , , and can be distinct.