Representation of algebraic distributive lattices with $\aleph\_1$ compact elements as ideal lattices of regular rings

Friedrich Wehrung

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Representation of algebraic distributive lattices with $\aleph\_1$ compact elements as ideal lattices of regular rings

Friedrich Wehrung

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Abstract

We prove the following result: Theorem. Every algebraic distributive lattice D with at most $\aleph\_1$ compact elements is isomorphic to the ideal lattice of a von Neumann regular ring R. (By earlier results of the author, the $\aleph\_1$ bound is optimal.) Therefore, D is also isomorphic to the congruence lattice of a sectionally complemented modular lattice L, namely, the principal right ideal lattice of R. Furthermore, if the largest element of D is compact, then one can assume that R is unital, respectively, that L has a largest element. This extends several known results of G.M. Bergman, A.P. Huhn, J. T˚uma, and of a joint work of G. Grätzer, H. Lakser, and the author, and it solves Problem 2 of the survey paper [10]. The main tool used in the proof of our result is an amalgamation theorem for semilattices and algebras (over a given division ring), a variant of previously known amalgamation theorems for semilattices and lattices, due to J. Tuma, and G. Grätzer, H. Lakser, and the author.

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Friedrich Wehrung. 2005-01-22. Representation of algebraic distributive lattices with $\aleph\_1$ compact elements as ideal lattices of regular rings. https://arxiv.org/abs/math/0501367

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