Routes and products of monoids

We investigate the problem of finding monoids that recognize languages of the form L-1 x L-T(2) where T is an arbitrary set of routes. We present a uniform method based on routes to find such monoids. Many classical operations from the theory of formal languages, such as catenation, bi-catenation, simple splicing, shuffle, literal shuffle, and insertion are shown to be just particular instances of the operation x(T).