Results on order decreasing full transformation semigroup via tropical geometry

Abstract

Let $\mathcal{D} \Upsilon_{\eta}$ be the subsemigroup of order decreasing full transformation on
${\Omega}_{\eta}= \left\{1,2,\cdots,\eta \right\}.$ In this paper, the tropical geometry were applied on
$\mathcal{D} \Upsilon_{\eta}$ for $1 \leq \eta \leq 4.$ The elements in $\mathcal{D} \Upsilon_{\eta}$ of
classical algebra were degenerated into tropical polynomial and some results were obtained through their
tropical curve.