Idempotent 2x2 matrices over linearly ordered abelian groups

In this paper we study the multiplicative semigroup of 2 × 2 matrices over a linearly ordered abelian group with an externally added bot tom element. The multiplication of such a semigroup is defined by replacing addition and multiplication by join and addition in the usual formula defining matrix multiplication. We show that there are four types of idempotents in such a matrix semigroup and we determine which of them are 0-primitive. We also prove that the poset of idempotents of such a matrix semigroup with respect to the natural order is a lattice. It turns out that such a matrix semigroup is inverse or orthodox if and only if the abelian group is trivial.