We generalize the construction of multi-tildes in the aim to provide double multitilde operators for regular languages. We show that the underlying algebraic structure involves the action of some operads. An operad is an algebraic structure that mimics the composition of the functions. The involved operads are described in terms of combinatorial objects. These operads are obtained from more pri...

We investigate finite torsors over big opens of spectra strongly $F$-regular germs that do not extend to the whole spectrum. Let $(R,\mathfrak{m},k)$ be a $k$-germ where $k$ is an algebraically closed field characteristic $p>0$. prove existence local cover $R \subset R^{\star}$ so $R^{\star}$ and: for all algebraic groups $G/k$ with solvable neutral component, every $G$-torsor open $\mathrm{Spe...

We explore the equimultiplicity theory of F-invariants Hilbert–Kunz multiplicity, F-signature, Frobenius Betti numbers, and Euler characteristic in strongly F-regular rings. Techniques introduced this paper provide a unified approach to study localization these invariants detection singularities.