(1 + 1 + 2)-Generated Equivalence Lattices,

Strietz [Proceedings of the Lattice Theory Conference, Ulm, 1975, pp. 257–259; Studia Sci. Math. Hungar.12 (1977), 1–17] and Zádori [Colloq. Math. Soc. János Bolyai43, “Lectures in Universal Algebra, Proceedings of the Conference, Szeged, 1983,” North-Holland, Amsterdam/New York, 1986, pp. 573–586] have shown that Equ(A), the lattice of all equivalences of a finite set A with A ≥ 7, has a four-element generating set such that exactly two of the generators are comparable. In other words, these lattices are (1 + 1 + 2)-generated. We extend this result for many infinite sets A; even for all sets if there are no inaccessible cardinals. Namely, we prove that if A is a set consisting of at least seven elements and there is no inaccessible cardinal ≤ A, then the complete lattice Equ(A) is (1 + 1 + 2)-generated. This result is sharp in the sense that Equ(A) has neither a three-element generating set nor a four-element generating set with more than one pair of comparable generators.