Idempotent 2x2 matrices over linearly ordered abelian groups

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.

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References

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Categories and General Algebraic Structures with Applications
Volume 21, Issue 1
July 2024
Pages 1-17

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