A presentation and some finiteness conditions for a new version of the Schützenberger product of monoids

In this paper we first define a new version of the Sch\"{u}tzenberger product for any two monoids A and B, and then, by defining a generating and relator set, we present some finite and infinite consequences of the main result. In the final part of this paper, we give necessary and sufficient conditions for this new version to be periodic and locally finite.