Max-Plus Algebra

Abstract

In max-plus algebra we work with the max-plus semi-ring which is the set ℝ_max=[-∞)∪ℝ together with operations 𝑎⊕𝑏 = max(𝑎,𝑏) and 𝑎⊗𝑏= 𝑎+𝑏. The additive and multiplicative identities are taken to be ε=-∞ and ε=0 respectively. Max-plus algebra is one of many idempotent semi-rings which have been considered in various fields of mathematics. Max-plus algebra is becoming more popular not only because its operations are associative, commutative and distributive as in conventional algebra but because it takes systems that are non-linear in conventional algebra and makes them linear. Max-plus algebra also arises as the algebra of asymptotic growth rates of functions in conventional algebra which will play a significant role in several aspects of this thesis. This thesis is a survey of max-plus algebra that will concentrate on max-plus linear algebra results. We will then consider from a max-plus perspective several results by Wentzell and Freidlin for finite state Markov chains with an asymptotic dependence.