Is the class of anticommutative magmas a variety?

Question

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?

Answer 1

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.