I have heard in universal algebra there is such a thing as a free variety, but is there such a thing as a free quasivariety? I would assume, that, for instance, in the language of a single binary operation symbol $*$, a free quasivariety is a free variety where additionally if $x*y=z*w$, then $x=z$ and $y=w$. Is this correct?
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There are no such concepts as "free variety" or "free quasi-variety" in Universal Algebra. There are free objects in a variety (quasi-variety).
markvs
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What is a free object in a quasi-variety? – user107952 Jul 10 '20 at 16:33
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The same definition as for variety. – markvs Jul 10 '20 at 16:38
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The thing is that if $\cal C$ is a set of algebras then the free algebras in the variety $var(\cal C)$ are in $qvar(\cal C)$. – markvs Jul 10 '20 at 16:41