Idempotent \(2\times 2\) matrices over linearly ordered abelian groups (Q6600770)

Tropical semirings and tropical matrices are interesting mathematical objects. \textit{M. Johnson} and \textit{M. Kambites} [Linear Algebra Appl. 435, No. 7, 1612--1625 (2011; Zbl 1228.15003)] studied the semigroup of of \(2 \times 2\) tropical matrices: their regularity, Green's relations, idempotent structure etc. In this article, the authors study the more general semigroup of \(2 \times 2\) matrices over a linearly ordered abelain group with an externally added bottom element. The authors provide a description of idempotents of this semigroup analogous to the \(2\times 2\) tropical semiring result. They show that the poset of idempotent matrices has the structure of a lattice. They also discuss the regularity of the semigroup.