k-theories and free inductive graded rings in abstract quadratic forms theories

We build on previous work on multirings ([17]) that provides generalizations of the available abstract quadratic forms theories (special groups and real semigroups) to the context of multirings ([10], [14]). 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 ([11]) and special groups k-theory, developed by dickmann-miraglia ([5]). we develop some properties of this generalized k-theory, that can be seen as a free inductive graded ring, a concept introduced in [2] in order to provide a solution of marshall’s signature conjecture.