Lattice theory and its applications

It seems appropriate, at a meeting called to celebrate G. Birkhoff's eightieth birthday, to lecture on a problem raised by him 45 years ago (see G. Birkhoff [1] and [2]), namely, the celebrated characterization problem of congruence lattices of algebras. In 1983, R. Wille raised the following closely connected question (see, e.g., K. Reuter and R. Wille [11]): Is every complete lattice L isomorphic to the lattice of complete congruence relations of a suitable complete lattice K '( S.-K. Teo [12] solved this problem in the finite case. At the 1988 Lisbon Meeting (see G. Gratzer [6]), the first author answered Wille's question in the affirmative. [6] also gave a detailed historical account of the problem. Since the Lisbon Meeting, a number of related results have been proved. G. Gratzer and H. Lakser [8] constructed a planar complete lattice Kj in [9] they noted that the lattice L of all m-complete congruence relations of an m-complete lattice K is malgebraic, and they proved a partial converse: Let m be a regular cardinal> No 1 and let L be an m-algebraic lattice with an m-compact unit element. Then L is isomorphic to the lattice of m-complete congruences of an m-complete lattice K. This contains the original result of G. Gratzer; indeed, if the lattice L is complete, then L is m-algebraic and every element of Lis m-compact provided that m is a regular cardinal satisfying m > ILl. A much sharper form of the original result of G; Gratzer was proved in the paper R. Freese, G. Gratzer, and E. T. Schmidt [3]: Every complete lattice L is isomorphic to the lattice of complete congruence relations of a complete modular lattice K. We combine and complete the two previous lines of investigations: