Document Type : Research Paper

**Authors**

Institute of Mathematics and Statistics, University of Tartu, Tartu, Estonia

**Abstract**

In 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.

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.

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.

**Keywords**

**Main Subjects**

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[5] Il’in, S.N., A regularity criterion for complete matrix semirings, Mat. Zametki 70(3) (2001), 366-374.

[6] Johnson, M. and Kambites, M., Multiplicative structure of 2 × 2 tropical matrices, Linear Algebra Appl. 435(7) (2011), 1612-1625.

[7] Weinert, H.J. and Wiegandt, R., On the structure of semifields and lattice-ordered groups, Period. Math. Hungar. 32(1-2) (1996), 129-147.