Abstract. We prove that there exists a lattice whose congruence lattice is not isomorphic to the congruence lattice of any lattice with $m$-permutable congruences. Our proof also extends to a wider class of algebras with $m$-permutable congruences. In order to do this we use and further develop the method invented by F. Wehrung for solving Dilworth's congruence lattice problem. To minimize the cardinality of our construction, we use the free trees combinatorial principle of P. Růžička.

AMS Subject Classification (1991): 06B10, 08A30, 06A12

Keyword(s): algebraic lattice, variety, congruence

Received March 6, 2007, and in revised form October 2, 2007. (Registered under 6000/2009.)