Tropical mathematics often is defined over an ordered cancellative
monoid $M$, usually taken to be $(\mathbb R, +)$ or $(\mathbb
Q, +)$. Although there is a rich theory arising from this
viewpoint, idempotent semirings possess a restricted algebraic
structure theory, and also do not reflect certain
valuation-theoretic properties, thereby forcing researchers to
rely often on combinatorial techniques.
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Over the last few years, often jointly with Izhakian and Knebusch,
we have studied
an alternative structure, more compatible with valuation theory, that permits
fuller use of algebraic structure in understanding the
underlying tropical geometry. The idempotent max-plus algebra $A$
of an ordered monoid $M$ is replaced by $R: = L\times M$,
where $L$ is a given indexing semiring (not necessarily with 0).
In this case we say $R$ is 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 $L = \{
\mathbb N \}$ we can describe multiple roots of polynomials via a
“layering function” $s: R \to L$.
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In this talk, we explain how supertropical algebras, and more generally layered
algebras, provide a robust algebraic foundation for tropical
algebraic geometry, in which the classical Zariski correspondence plays a major role.
We also introduce a new approach of Perri involving homomorphisms of ordered groups.