Geometric and conditional geometric equivalences of algebras (Q354473)

Two universal algebras are said to be geometrically equivalent (conditionally geometrically equivalent) if pairs of different systems of termal equations (of quantifier-free elementary formulas) defining common algebraic sets (definable subsets) are properly the same. An algebra \(A\) is called uniformly locally finite if there exists a function \(h:\omega\to\omega\) such that for any \(n\in\omega\) and any \(B\subseteq A\) the inequality \(|B|\leq n\) entails for the subalgebra \((B)_A\) of \(A\) generated by the set \(B\) the equality \(|(B)_A|\leq h(n)\). The main result of the paper states that for any uniformly locally finite algebras \(A_1\) and \(A_2\) of finite signature the following conditions are equivalent: (a) \(A_1\) and \(A_2\) are geometrically equivalent; (b) for any infinite Boolean algebra \(B\) the \(B\)-powers of \(A_1\) and \(A_2\) are conditionally geometrically equivalent.