Unified-like product of monoids and its regularity property

We first define a new monoid construction (called unified-like product O ◊_Ω J) under a unified product O ⋈ J and the Schützenberger product O ◊ J. We investigate whether this algebraic construction defined with operations of the unified and Schützenberger 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 ◊_Ω J to be regular.