2005/01/22 by Friedrich Wehrung
Mathematics · #math.GM #math.RA #msc:16E50 #msc:16D25 #msc:06A12 #msc:06C20
published as Publicacions Matematiques (Barcelona) 44, no. 2 (2000) 419--435
arxiv created 2005/01/22 · arxiv updated 2009/12/01
We prove the following result: Theorem. Every algebraic distributive lattice D with at most ℵ_1 compact elements is isomorphic to the ideal lattice of a von Neumann regular ring R. (By earlier results of the author, the ℵ_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.