Abstract
This chapter studies \(\mathcal{R}\)-trivial monoids, that is, monoids where Green’s relation \(\mathcal{R}\) is the equality relation. They form an important class of finite monoids, which have quite recently found applications in the analysis of Markov chains; see Chapter 14 The first section of this chapter discusses lattices and prime ideals of arbitrary finite monoids, as they shall play an important role in both the general theory and in the structure theory of \(\mathcal{R}\)-trivial monoids. The second section concerns \(\mathcal{R}\)-trivial monoids and, in particular, left regular bands, which are precisely the regular \(\mathcal{R}\)-trivial monoids.
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Steinberg, B. (2016). 2 \(\mathcal{R}\)-trivial Monoids. In: Representation Theory of Finite Monoids. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-43932-7_2
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