Fix an algebraic signature $\Omega$. Let $F$ be a finite set of equations in $\Omega$. Is it decidable if the set $F$ has only trivial models? By trivial models, I mean one-element models. For example, if the set $F$ contains the equation $x=y$ then certainly it has only trivial models. However, perhaps the set $F$ does not contain $x=y$ but it still has only trivial models. I am wondering if the problem is decidable either way.
2026-02-24 05:10:28.1771909828
Is it decidable if a finite set of equations have only trivial models?
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No.
Perkins, Peter
Unsolvable problems for equational theories.
Notre Dame J. Formal Logic 8 (1967), 175-185.
states
Theorem 14. There is no effective method for determining whether or not an arbitrary finite set of equations in one binary operation symbol and no constants is consistent.