Two examples from lattice theory: is every lattice with unique complements distributive? [no] is every distributive algebraic lattice isomorphic to the lattice of congruences of a lattice? [no] See http://www.ams.org/notices/200706/tx070600696p.pdf
Grätzer, George, Two problems that shaped a century of lattice theory, Notices Am. Math. Soc. 54, No. 6, 696-707 (2007). ZBL1286.06001, MR2327971.
