F-singularities of pairs and Inversion of Adjunction of arbitrary codimension

We generalize the notions of F-regular and F-pure rings to pairs $(R,{\text{a}}^t)$ of rings $R$ and ideals $\text{a}\subset R$ with real exponent $t>0$, and investigate these properties. These ``F-singularities of pairs’’ correspond to singularities of pairs of arbitrary codimension in birational geometry. Via this correspondence, we prove Inversion of Adjunction of arbitrary codimension, which states that for a pair $(X,Y)$ of a smooth variety $X$ and a closed subscheme $Y⊊X$, if the restriction $(Z,Y{\mathrm{\mid }}_Z)$ to a normal $\text{Q}$-Gorenstein closed subvariety $Z⊊X$ is klt (resp. lc), then the pair $(X,Y+Z)$ is plt (resp. lc) near $Z$.