Abstract
In tropical mathematics, as well as other mathematical theories involving semirings, one often is challenged by the lack of negation when trying to formulate the tropical versions of classical algebraic concepts for which the negative is a crucial ingredient. Following ideas originating in work of Dress, Gaubert, and the Max-Plus group and pursued further by Akian, Gaubert, and Guterman, we study algebraic structures with negation maps, called systems, showing how these unify the more viable (super)tropical versions, as well as symmetrization, hypergroup theory and fuzzy rings, thereby helping to explain similarities in these theories. Special attention is paid to metatangible systems, whose algebraic theory includes all the main examples, and is rich enough to facilitate computations and provide a host of structural results. The systems studied in this paper are “ground” systems, insofar as they are the underlying structure which can be studied via other “systemic modules”. By formalizing the structure, we can introduce morphisms. Morphisms enable us to describe the tropicalization functor, as well as providing a link between classical algebraic results and their tropical and hyperfield analogs.
| Original language | English |
|---|---|
| Pages (from-to) | 62-138 |
| Number of pages | 77 |
| Journal | European Journal of Mathematics |
| Volume | 8 |
| Issue number | 1 |
| DOIs | |
| State | Published - Mar 2022 |
Bibliographical note
Publisher Copyright:© 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
Funding
The author would like to thank the following researchers for helpful conversations: Oliver Lorscheid and Zur Izhakian for discussions on tropicalization, Marianne Akian and Stephane Gaubert, together with Adi Niv, for discussions on symmetrization and for interesting examples, Max Knebusch for insights on ub semirings, Matt Baker for explaining hyperfields, and especially the role of the “sign hyperfield,” and Sergio Lopez for helpful comments on the first arXiv version. Special thanks are due to Guy Shachar for a careful reading of the first arXiv version which led to several corrections, to J. Jun for finding several inaccuracies in various versions, pointing out the connection to fuzzy rings, Henry’s paper, and other interesting papers with which he has been involved, to Marianne Akian for pointing out how various theorems should be clarified, and especially to the referee, who provided a great many improvements. The author’s research was supported by Israel Science Foundation grants No. 1207/12 and 1994/20. The author would also like to thank the University of Virginia for its support during the initial preparation of this work in 2015.
| Funders | Funder number |
|---|---|
| Guy Shachar | |
| Max Knebusch | |
| Israel Science Foundation | 1207/12, 1994/20 |
Keywords
- Congruence
- ELT algebra
- Exploded algebra
- Exterior algebra
- Fuzzy ring
- Grassmann algebra
- Hyperfield
- Lie algebra
- Linear algebra
- Matrix
- Metatangible
- Module
- Monoid
- Negation map
- Polynomial
- Puiseux series
- Semifield
- Semigroup
- Semiring
- Superalgebra
- Supertropical algebra
- Surpassing relation
- Symmetrization
- System
- Tangible
- Tensor product
- Triple
- Tropical algebra
- Tropical geometry
- Tropicalization
- Valuation