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- W1639175025 abstract "Tropical mathematics often is defined over an ordered cancellative monoid $tM$, usually taken to be $(RR, +)$ or $(QQ, +)$. Although a rich theory has arisen from this viewpoint, cf. [L1], idempotent semirings possess a restricted algebraic structure theory, and also do not reflect certain valuation-theoretic properties, thereby forcing researchers to rely often on combinatoric techniques. In this paper we describe an alternative structure, more compatible with valuation theory, studied by the authors over the past few years, that permits fuller use of algebraic theory especially in understanding the underlying tropical geometry. The idempotent max-plus algebra $A$ of an ordered monoid $tM$ is replaced by $R: = Ltimes tM$, where $L$ is a given indexing semiring (not necessarily with 0). In this case we say $R$ layered by $L$. When $L$ is trivial, i.e, $L={1}$, $R$ is the usual bipotent max-plus algebra. When $L={1,infty}$ we recover the standard supertropical structure with its ghost layer. When $L = NN $ we can describe multiple roots of polynomials via a function $s: R to L$. Likewise, one can define the layering $s: R^{(n)} to L^{(n)}$ componentwise; vectors $v_1, dots, v_m$ are called tropically dependent if each component of some nontrivial linear combination $sum a_i v_i$ is a ghost, for tangible $a_i in R$. Then an $ntimes n$ matrix has tropically dependent rows iff its permanent is a ghost. We explain how supertropical algebras, and more generally layered algebras, provide a robust algebraic foundation for tropical linear algebra, in which many classical tools are available. In the process, we provide some new results concerning the rank of d-independent sets (such as the fact that they are semi-additive),put them in the context of supertropical bilinear forms, and lay the matrix theory in the framework of identities of semirings." @default.
- W1639175025 created "2016-06-24" @default.
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- W1639175025 date "2014-01-01" @default.
- W1639175025 modified "2023-09-24" @default.
- W1639175025 title "Algebraic structures of tropical mathematics" @default.
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- W1639175025 doi "https://doi.org/10.1090/conm/616/12312" @default.
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