74 research outputs found
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
Hopf algebras for matroids over hyperfields
Recently, M.~Baker and N.~Bowler introduced the notion of matroids over hyperfields as a unifying theory of various generalizations of matroids. In this paper we generalize the notion of minors and direct sums from ordinary matroids to matroids over hyperfields. Using this we generalize the classical construction of matroid-minor Hopf algebras to the case of matroids over hyperfields
On Fuzzy Gamma Hypermodules
Let R be a Γ-hyperring and M be an Γ -hypermodule over R. We introduce and tudy fuzzy RΓ -hypermodules. Also, we associate a Γ- hypermodule to every fuzzy Γ-hypermodule and investigate its basic properties
History and new possible research directions of hyperstructures
We present a summary of the origins and current developments of the theory of algebraic hyperstructures. We also sketch some possible lines of research
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