In this joint work with Pawel Gladki (University of Silesia, Poland) and Kaique M.A. Roberto (Faculty Einstein, Brazil), we develop a new approach to local–global principles in (non-reduced) algebraic theory of quadratic forms by employing instruments and techniques of hyper compositional algebra (a.k.a. multi-algebras).

We define quadratic extensions of special hyperfields (hyperfields that faithfully encode quadratic forms theory) as a particular type of Marshall quotients and investigate their basic properties. In particular we apply these constructions to develop new types of local–global principles in the algebraic theory of quadratic forms that we call n-th Pfister approximation properties (n-PAP) and analyze a few instances where they do—as well as do not—hold. This approach builds on Pfister's and Marshall's local–global principles that have been established in the realm of reduced theories of quadratic forms.

References:
[1] Pawel Gladki, Krzysztof Worytkiewicz. "Witt rings of quadratically presentable fields". Categories and General Algebraic Structures 12(1): 1-23, 2020.
[2] Hugo R. O. Ribeiro, Kaique M. A. Roberto, Hugo L. Mariano. "Functorial relationship between multirings and the various abstract theories of quadratic forms". São Paulo Journal of Mathematical Sciences 16(1):5–42, 2022.
[3] Kaique M. A. Roberto, Hugo R. O. Ribeiro, Hugo L. Mariano. "Quadratic structures associated to (multi) rings". Categories and General Algebraic Structures 16(1):105–141, 2022.