K-theories and Free Inductive Graded Rings in Abstract Quadratic Forms Theories

We build on previous work on multirings ({roberto2021quadratic}) that provides generalizations of the available abstract quadratic forms theories (special groups and real semigroups) to the context of multirings ({marshall2006real}, {ribeiro2016functorial}). Here we raise one step in this generalization, introducing the concept of pre-special hyperfields and expand a fundamental tool in quadratic forms theory to the more general multivalued setting: the K-theory. We introduce and develop the K-theory of hyperbolic hyperfields that generalize simultaneously Milnor's K-theory ({milnor1970algebraick}) and Special Groups K-theory, developed by Dickmann-Miraglia ({dickmann2006algebraic}). We develop some properties of this generalized K-theory, that can be seen as a free inductive graded ring, a concept introduced in {dickmann1998quadratic} in order to provide a solution of Marshall's Signature Conjecture.