Matroids over tracts provide an algebraic framework simultaneously generalizing the notions of matroids, oriented and valuated presented by Baker Bowler. Pendavingh partially extended this theory to skew hyperfields a new axiom system in terms quasi-Pl\"ucker coordinates. We present matroids tracts, which generalizes both weak developed Pendavingh. give several cryptomorphic systems for such ci...

In this paper, we develop the rudiments of a tropical homology theory. We use language “triples” and “systems” to simultaneously treat structures from various approaches mathematics, including semirings, hyperfields, super algebra. enrich algebraic with negation map where it does not exist naturally. obtain an analogue Schanuel’s lemma which allows us talk about projective dimension modules in ...

We define the universal 1-adic thickening of the field of real numbers. This construction is performed in three steps which parallel the universal perfection, the Witt construction and a completion process. We show that the transposition of the perfection process at the real archimedean place is identical to the “dequantization” process and yields Viro’s tropical real hyperfield T R. Then we pr...

Let $G$ be a monoid with identity $e$. In this paper, first we introduce the notions of $G$-graded hyperrings, graded hyperideals and graded hyperfields in the sense of Krasner hyperring $R$. Also, we define the notion of a greded $R$-hypermodules and some examples are presented. Then we investigate graded maximal, graded prime and graded primary hyperideals of a graded hyperring $R$. Finally, ...

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 hy...

We define the universal thickening of the field of real numbers. This construction is performed in three steps which parallel the universal perfection, the Witt construction and a completion process. We show that the transposition of the perfection process at the real archimedean place is identical to the “dequantization” process and yields Viro’s tropical real hyperfield R[. Then, we prove tha...

This paper introduces an approach to the axiomatic theory of quadratic forms based on {tmem{presentable}} partially ordered sets, that is partially ordered sets subject to additional conditions which amount to a strong form of local presentability. It turns out that the classical notion of the Witt ring of symmetric bilinear forms over a field makes sense in the context of {tmem{quadratically p...

In [3], Nathan Bowler and the first author introduced a category of algebraic objects called tracts defined notion (weak strong) matroids over tract. part paper, we summarize clarify connections to other which have previously been used in connection with matroid theory. For example, show that both partial fields hyperfields are fuzzy rings, rings tracts, these relations compatible theories. We ...