Consistent pairs of $\mathfrak{s}$-torsion pairs in extriangulated categories with negative first extensions

As a generalization of a consistent pair of $t$-structures on triangulated categories, we introduced the notion of a consistent pair of $\mathfrak{s}$-torsion pairs in the extriangulated setup. Let $({\mathcal{T}}_i,{\mathcal{F}}_i)$ be an $\mathfrak{s}$-torsion pair in an extriangulated category with a negative first extension for any $i=1,2$. By using the consistent pair, we gave a criterion for $({\mathcal{T}}_1\ast {\mathcal{T}}_2,{\mathcal{F}}_1\cap {\mathcal{F}}_2)$ to be an $\mathfrak{s}$-torsion pair. Our results were then applied to the torsion theory induced by $\tau$-rigid modules.