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This is a follow-up to my previous question, here: Is it decidable if a finite set of equations have only trivial models?. Let our signature be that of a single binary operation symbol $*$. Suppose I am given a finite set $S$ of identities in that signature, and I want to know whether the set $S$ implies the commutative identity $x*y=y*x$. Is that a decidable problem?

user107952
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    More generally, it would be interesting to know which (if any) equations $E$ are "decidably implied" by finite equational theories. I suspect that no such equations exist at all, but I don't immediately see the proof. – Noah Schweber Aug 16 '23 at 20:10
  • @NoahSchweber Well, except for the trivial equation $t=t$, which is always implied by any set of equations.. – user107952 Aug 18 '23 at 01:07

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