Endomorphisms of the semigroup of nonnegative invertible matrices of order two over commutative ordered rings

Let $R$ be a linearly ordered commutative ring with $1/2$ generated by its invertible elements, $G_2(R)$ be the subsemigroup in $\mathrm{GL}_2(R)$ consisting of all matrices with nonnegative elements. In this paper, we describe endomorphisms of the given semigroup.