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

We build on previous work multirings ([17]) that providesgeneralizations of the available abstract quadratic forms theories (specialgroups and real semigroups) to context ([10], [14]). Herewe raise one step in this generalization, introducing concept pre-specialhyperfields expand a fundamental tool theory themore general multivalued setting: K-theory. introduce developthe K-theory hyperbolic hyperfields generalize simultaneously Milnor’sK-theory ([11]) Special Groups K-theory, developed by Dickmann-Miraglia ([5]). develop some properties generalized thatcan be seen as free inductive graded ring, introduced [2] inorder provide solution Marshall’s Signature Conjecture.