In a previous question, I was told that free algebras on sufficiently many generators are generic algebras. That raises the question, what is an example of a free algebra that is not generic? An algebra $A$ is generic for a variety $V$ iff $A$ generates $V$.
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In what sense does it generate $V$? As the least variety containing it? – Jakobian Jan 30 '21 at 22:36
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5For example the free group on one generator is surely not generic among the groups because it's commutative. – Berci Jan 30 '21 at 22:42
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@Berci You can post that as an answer. – user107952 Jan 30 '21 at 23:55
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For example the free group [or free monoid] on one generator is surely not generic among all the groups [monoids], because it's commutative.
Berci
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This question is somewhat related to this MO question. The MO question asks for examples of locally finite varieties that are not finitely generated. If a variety is locally finite, then its finitely generated free algebras are finite. If the variety is not finitely generated, then no finite algebra in the variety is generic. So the examples given at the link are of varieties where no free algebra of finite rank is generic.
Keith Kearnes
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