On minimal ideals in pseudo-finite semigroups
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A semigroup S is said to be right pseudo-finite if the universal right congruence can be generated by a finite set U⊆S×S, and there is a bound on the length of derivations for an arbitrary pair (s,t)∈S×S as a consequence of those in U. This article explores the existence and nature of a minimal ideal in a right pseudo-finite semigroup. Continuing the theme started in an earlier work by Dandan et al., we show that in several natural classes of monoids, right pseudo-finiteness implies the existence of a completely simple minimal ideal. This is the case for orthodox monoids, completely regular monoids and right reversible monoids, which include all commutative monoids. We also show that certain other conditions imply the existence of a minimal ideal, which need not be completely simple; notably, this is the case for semigroups in which one of the Green's pre-orders ≤L or ≤J is left compatible with multiplication. Finally, we establish a number of examples of pseudo-finite monoids without a minimal ideal. We develop an explicit construction that yields such examples with additional desired properties, for instance, regularity or J-triviality.
Gould , V , Miller , C , Quinn-Gregson , T & Ruskuc , N 2022 , ' On minimal ideals in pseudo-finite semigroups ' , Canadian Journal of Mathematics , vol. FirstView . https://doi.org/10.4153/S0008414X2200061X
Canadian Journal of Mathematics
Copyright © 2022 Published by Cambridge University Press on behalf of The Canadian Mathematical Society. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.4153/S0008414X2200061X.Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
DescriptionFunding: This work was supported by the Engineering and Physical Sciences Research Council [EP/V002953/1].
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