The max-plus semiring:$(\{\mathbb{R}\cup{-\infty}\}, max,+):=(\{\mathbb{R}\cup{-\infty}\}, \oplus, \otimes)$ is defined as an algebraic structure that has max as addition and + as multiplication.
My question is what "argmax" means in this semiring or algebraic structure.
I have never seen the explanation.
Common addition doesn't have accompanied structure like "argmax". But I think "argmax" is not only a necessary accompanied structure of the max-plus semiring but also important information we often use.
Does anyone explain how should we understand argmax from algebraic viewpoint?