Artinianness and finiteness of formal local cohomology modules with respect to a pair of ideals

Let \((R,\mathfrak {m})\) be a commutative Noetherian local ring, M be a finitely generated R-module and \(\mathfrak {a}\), I and J are ideals of R. We investigate the structure of formal local cohomology modules of \(\mathfrak {F}^i_{\mathfrak {a},I,J}(M)\) and \(\check{\mathfrak {F}}^i_{\mathfrak {a},I,J}(M)\) with respect to a pair of ideals, for all \(i\ge 0\). The main subject of the paper is to study the finiteness properties and artinianness of \(\mathfrak {F}^i_{\mathfrak {a},I,J}(M)\) and \(\check{\mathfrak {F}}^i_{\mathfrak {a},\mathfrak {m},J}(M)\). We study the maximum and minimum integer \(i\in \mathbb {N}\) such that \(\mathfrak {F}^i_{\mathfrak {a},\mathfrak {m},J}(M)\) and \(\check{\mathfrak {F}}^i_{\mathfrak {a},\mathfrak {m},J}(M)\) are not Artinian and we obtain some results involving cosupport, coassociated and attached primes for formal local cohomology modules with respect to a pair of ideals. Also, we give an criterion involving the concepts of finiteness and vanishing of formal local cohomology modules and Cech-formal local cohomology modules with respect to a pair of ideals.