If a semigroup S has no proper ideals can it have both regular and non-regular members? My guess would be 'yes' but in that case does anyone know of an example in the literature?
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Yes. In section 8.5 of volume 2 of Clifford and Preston they give a construction due to Bruck that embeds any monoid $M$ is a simple monoid $C(M)$. In Thm 8.48 they say $C(M)$ is regular iff $M$ is regular. So taking any nonregular monoid $M$ gives you $C(M)$ as the answer.
answered Mar 22, 2019 at 0:48