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

Document Type : Research Paper


Instituto de Matemática e Estatística, Universidade de Sao Paulo, Brazil.


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.


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