Is there any classification of countably infinite monoids with minimal right ideal? or at least in some classes of monoids?
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$\begingroup$ What would you hope to mean by a classifications? You can take any countable monoid and adjoin a zero to get a minimal right ideal. $\endgroup$– Benjamin SteinbergCommented Jul 13, 2022 at 13:01
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1$\begingroup$ And all finite or compact monoids have minimal right ideals. $\endgroup$– Benjamin SteinbergCommented Jul 13, 2022 at 13:01
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$\begingroup$ @Benjamin Steinberg Thanks for the guidance! For example, all commutative artinian monoids have this property. I want to know the other classes of monoids with this property and you kindly introduced me two new families. $\endgroup$– khersCommented Jul 13, 2022 at 14:25
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1$\begingroup$ Asking for classes with this property is more hopeful than asking for a classification. Any monoid with descending chain condition on right ideals (e.g. finite monoids) has it, but there are many other ways to get them like artificially adding a 0. $\endgroup$– Benjamin SteinbergCommented Jul 13, 2022 at 17:30
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1$\begingroup$ I think you can find some stuff on this in Clifford and Preston's classic book on semigroup theory. They call it the minimum condition on right ideals. $\endgroup$– Benjamin SteinbergCommented Jul 13, 2022 at 19:10
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