General formal local cohomology modules

Let $(R,\mathfrak{m})$ be a local ring, $\Phi$ a system of ideals of $R$ and $M$ a finitely generated $R$-module. In this paper, we define and study general formal local cohomology modules. We denote the $i$-th general formal local cohomology module $M$ with respect to $\Phi$ by $\mathfrak{F}_{\Phi}^{i}(M)$ and we investigate some finiteness and Artinianness properties of general formal local cohomology modules.