Free products of operads and Gröbner base of some operads

In this paper, we provide an explicit definition of the free product of operads and present a method for determining its basis and dimension. We demonstrate the application of these tools to well-known operads that have two operations. Additionally, we provide examples of operads that are isomorphic to the free product of operads that arise from commutators and anti-commutators. Namely, these operads are Lie-admissible, associative admissible, and metabelian-admissible. Furthermore, we introduce an operad that generalizes the Leibniz operad and prove that an anti-commutative operad can be embedded into this operad via a commutator. We also construct a basis for this operad.