Generalized factorials characterized by Dirichlet convolution

We extend A.B. Mingarelli’s method for constructing generalized factorials. Our extension uses a pair of arithmetic functions $(x,y)$, where $x$ is superadditive. When $x$ is the identity function, our generalized factorial reduces to Mingarelli’s. A result on the irrationality of the Euler constant within this framework is given. Using Dirichlet convolution, we characterize when two pairs $(\alpha ,\beta )$ and $(x,y)$ generate the same factorials.