82 research outputs found
From monoids to hyperstructures: in search of an absolute arithmetic
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
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 , the functor taking a pasture
to the set of isomorphism classes of rank- -matroids on is
representable by an ordered blue scheme , the moduli space of
rank- matroids on .
In the third part, we draw conclusions on matroid theory. A classical
rank- matroid on corresponds to a -valued point of
where is the Krasner hyperfield. Such a point defines a
residue pasture , which we call the universal pasture of . We show that
for every pasture , morphisms are canonically in bijection with
-matroid structures on .
An analogous weak universal pasture classifies weak -matroid
structures on . The unit group of can be canonically identified with
the Tutte group of . We call the sub-pasture of generated by
``cross-ratios' the foundation of ,. It parametrizes rescaling classes of
weak -matroid structures on , and its unit group is coincides with the
inner Tutte group of . We show that a matroid is regular if and only if
its foundation is the regular partial field, and a non-regular matroid 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
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
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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