Varieties of initial dialgebras and some of their Koszul dual operads

In this paper, for a given variety $\text{Var}$, we present a universal algorithm for constructing a subvariety of $\text{Var}$-dialgebras from which one can recover an algebra belonging to $\text{Var}$. Such a subvariety is called the variety of initial $\text{Var}$-dialgebras. In addition, we construct a basis of the free initial Lie and associative dialgebras.