Algebraic structures of tropical mathematics
Zur Izhakian, Manfred Knebusch, Louis Halle Rowen · Contemporary mathematics - American Mathematical Society · 2014
Tropical mathematics often is defined over an ordered cancellative monoid M \mathcal {M} , usually taken to be ( R , + ) (\mathbb R, +) or ( Q , + ) (\mathbb Q, +) . Although a rich theory has arisen from this viewpoint (cf. G.L. Litvinov, The Maslov dequantization, and idempotent and tropical mathematics; a brief introduction , 2005), 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 A of an ordered monoid M \mathcal {M} is replaced by R := L × M R: = L\times \mathcal {M} , where L L is a given indexing semiring (not necessarily with 0). In this case we say R R layered by L L . When L L is trivial, i.e., L = { 1 } L = \{ 1 \} , R R is the usual bipotent max-plus algebra. When L = { 1 , ∞ } L = \{ 1, \infty \}