Unified-like product of monoids and its regularity property

We first define a new monoid construction (called unified-like product O (sic)(Omega)J) under a unified product O (sic) J and the Schutzenberger product O (sic) J. We investigate whether this algebraic construction defined with operations of the unified and Schutzenberger product specifies a monoid or not. Then, we obtain a presentation of this new product for any two monoids. Finally, we define the necessary and sufficient conditions for O (sic)(Omega)J to be regular.