82 research outputs found

    From monoids to hyperstructures: in search of an absolute arithmetic

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    We show that the trace formula interpretation of the explicit formulas expresses the counting function N(q) of the hypothetical curve C associated to the Riemann zeta function, as an intersection number involving the scaling action on the adele class space. Then, we discuss the algebraic structure of the adele class space both as a monoid and as a hyperring. We construct an extension R^{convex} of the hyperfield S of signs, which is the hyperfield analogue of the semifield R_+^{max} of tropical geometry, admitting a one parameter group of automorphisms fixing S. Finally, we develop function theory over Spec(S) and we show how to recover the field of real numbers from a purely algebraic construction, as the function theory over Spec(S).Comment: 43 pages, 1 figur

    The moduli space of matroids

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    In the first part of the paper, we clarify the connections between several algebraic objects appearing in matroid theory: both partial fields and hyperfields are fuzzy rings, fuzzy rings are tracts, and these relations are compatible with the respective matroid theories. Moreover, fuzzy rings are ordered blueprints and lie in the intersection of tracts with ordered blueprints; we call the objects of this intersection pastures. In the second part, we construct moduli spaces for matroids over pastures. We show that, for any non-empty finite set EE, the functor taking a pasture FF to the set of isomorphism classes of rank-rr FF-matroids on EE is representable by an ordered blue scheme Mat(r,E)Mat(r,E), the moduli space of rank-rr matroids on EE. In the third part, we draw conclusions on matroid theory. A classical rank-rr matroid MM on EE corresponds to a K\mathbb{K}-valued point of Mat(r,E)Mat(r,E) where K\mathbb{K} is the Krasner hyperfield. Such a point defines a residue pasture kMk_M, which we call the universal pasture of MM. We show that for every pasture FF, morphisms kM→Fk_M\to F are canonically in bijection with FF-matroid structures on MM. An analogous weak universal pasture kMwk_M^w classifies weak FF-matroid structures on MM. The unit group of kMwk_M^w can be canonically identified with the Tutte group of MM. We call the sub-pasture kMfk_M^f of kMwk_M^w generated by ``cross-ratios' the foundation of MM,. It parametrizes rescaling classes of weak FF-matroid structures on MM, and its unit group is coincides with the inner Tutte group of MM. We show that a matroid MM is regular if and only if its foundation is the regular partial field, and a non-regular matroid MM is binary if and only if its foundation is the field with two elements. This yields a new proof of the fact that a matroid is regular if and only if it is both binary and orientable.Comment: 83 page

    (weakly) (s,n)-closed hyperideals

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    A multiplicative hyperring is a well-known type of algebraic hyperstructures which extend a ring to a structure in which the addition is an operation but multiplication is a hyperoperation. Let G be a commutative multiplicative hyperring and s,n \in Z^+. A proper hyperideal Q of G is called (weakly) (s,n)-closed if (0 \neq a^s \subseteq Q) s^s \subseteq Q for a\in G implies a^n \subseteq Q. In this paper, we aim to investigate (weakly) (s,n)-closed hyperideals and give some results explaining the structures of these notions

    On Fuzzy Gamma Hypermodules

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    Let R be a Ξ“-hyperring and M be an Ξ“ -hypermodule over R. We introduce and tudy fuzzy RΞ“ -hypermodules. Also, we associate a Ξ“- hypermodule to every fuzzy Ξ“-hypermodule and investigate its basic properties
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