A generalization of bicrossed product of monoids

In this paper, we define [Formula: see text]-generalized Schützenberger-bicrossed product from the point of Combinatorial Group Theory. We give to be a monoid of this new product, then we obtain a presentation for [Formula: see text]-generalized Schützenberger-bicrossed product of arbitrary monoids. Additionally, we give the necessary and sufficient conditions for the regularity of this product.