A magma is simply a set $S$ with a single binary operation $*$. An anticommutative magma, under my definition, is one where $x*y=y*x \implies x=y$. This is certainly a quasivariety, simply by definition. Is it in fact a variety?
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No, the class of anticommutative magmas is not a variety.
Take the free magma on two generators $x$ and $y$. This magma is clearly anticommutative. If one identifies $xy$ with $yx$ in this magma, the quotient is not anticommutative anymore.
Geoffrey Trang
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