Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-5853Articles in Press20231030Idempotent 2x2 matrices over linearly ordered abelian groups10400110.48308/cgasa.2023.232266.1412ENValdis LaanInstitute of Mathematics and Statistics, University of Tartu, Tartu, EstoniaMarilyn KuttiInstitute of Mathematics and Statistics, University of Tartu, Tartu, EstoniaJournal Article20230704In this paper we study multiplicative semigroups of $2\times 2$ matrices over a linearly ordered abelian group with an externally added bottom 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 this semigroup and we determine which of them are $0$-primitive. <br />We also prove that the poset of idempotents with respect to the natural order is a lattice. It turns out that this matrix semigroup is inverse or orthodox if and only if the abelian group is trivial.https://cgasa.sbu.ac.ir/article_104001_6f3bc63a18e0e31a141fe9046f5f5e2e.pdf