Green’s relations for 2 × 2 matrices over linearly ordered abelian groups

We consider semigroups of 2 × 2 matrices over linearly ordered abelian groups with respect to multiplication, which is defined similarly to tropical algebra. We study Green’s relations on such semigroups. In particular, we describe the R-, L- and H-classes of such semigroups and give a simple criterion for determining whether two matrices are D-related. We prove that the D-relation coincides with the J-relation. We also study maximal subgroups of such semigroups. It turns out that if the abelian group is divisible, then these maximal subgroups can have two different forms.