G. Gratzer and E. T. Schmidt: m-complete congruence lattices. 93. For an m-complete lattice V and an m-complete congruence a on V, we define the prime.
Lattice Theory and its Applications K.A. Baker, R. Wille (OOs.) © Heldermann Verlag 1995 91-101
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Algebraic lattices as congruence lattices: The m-complete case G. Gratzer *
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E. T. Schmidt f
Introduction
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:
Theorem Let m be a regular cardinal> No. Every m-algebraic lattice L is isomorphic to the lattice of m-complete congruence relations of a suitable m-complete modular lattice K. We construct the lattice K by combining the techniques developed in G. Gratzer and H. Lakser [8] and R. Freese, G. Gratzer, and E. T. Schmidt [3]. In this lecture, we give the construction of K, and outline the proof of the Theorem as a series of lemmas stated without proofs. The reader should not have much difficulty supplying the computations that constitute the proofs. The same construction with a completely different proof will appear in G. Gratzer, P. Johnson, and E. T. Schmidt [10]. ·Research supported by the NSERC of Canada t Research supported by the Hungarian National Foundation for Scientific Research, under Grant No. 1003
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G. Gratzer and E. T, Schmidt: m-complete congruence lattices
In this J.C,",IJUJ.'C;, m always stands for an infinite L is the m-
