On definability of the equality in classes of algebras with an equivalence relation
In an earlier paper, the author has presented [``A finitary 1-equivalential logic not finitely equivalential'', Bull. Sect. Log., Univ. Lódź, Dep. Log. 24, No. 3, 120-123 (1995; Zbl 0841.03037)] a (finitary) regularly algebraizable logic in the similarity type \(\{\leftrightarrow\}\) that is not finitely equivalential. It is now shown that such logics exist in every similarity type containing \(\leftrightarrow\). As a consequence, the following model-theoretic result concerning algebras with an equivalence relation is obtained: for any algebraic similarity type, there are classes of such algebras in which equality is definable by a set of atomic formulas, but not by a finite set of such formulas.
scientific article; zbMATH DE number 1501386