On upper continuous and semimodular lattices

Four well known semimodularity conditions are considered: Birkhoff condition, upper semimodularity, MacLane condition,M-symmetry. It is shown that, for upper continuous strongly atomic lattices, these conditions are equivalent. It is also shown that, for lower continuous strongly coatomic lattices,M-symmetry and upper semimodularity coincide, but are different from the MacLane condition; this is so even for lower continuous lattices with ACC. An example of a nonalgebraic upper continuous strongly atomic lattice is given.Presented by R. Freese.