Logics with certain generalized quantifiers are constructed in this paper. The introduced quantifiers allow to study lattices of congruences and lattices of subalgebras. The formulas of the languages \(L_{\omega\omega}(\approx)\) and \(L_{\omega\omega}(\langle\;\;\rangle)\) are inductively defined and both have the Skolem-Löwenheim property. The Hanf number of \(L_{\omega\omega}(\approx)\) is equal to the Hanf number of \(L_{\omega\omega}(\langle\;\;\rangle)\). Both languages are neither compact nor axiomatizable. The theory of a finitely generated discriminator variety is decidable in the language \(L_{\omega\omega}(\approx)\), but the theory of a non-locally finite variety of Abelian groups is undecidable in \(L_{\omega\omega}(\approx)\) as well as in \(L_{\omega\omega}(\langle\;\;\rangle)\). Finally, the author mentions similar results on the languages \(L_{\omega\omega}({\mathbf A})\) and \(L_{\omega\omega}(\equiv)\) concerning the automorphism quantifier \({\mathbf A}\) and the quantifier \(\equiv\) of elementary equivalence, respectively.
For the entire collection see [Zbl 0885.00035].