2000/07/01 by Friedrich Wehrung · 1 citation
Computer Science · Mathematics · Decision Sciences · #Advanced Algebra and Logic #Rings, Modules, and Algebras #Fuzzy and Soft Set Theory #Mathematics #Distributive property #Ideal (ethics) #Aleph #Algebraic number #Representation (politics) #Pure mathematics #Algebra over a field #Distributive lattice #Mathematical analysis #Physics
paper · doi:10.5565/publmat_44200_03
openalex publication_date 2000/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/02
We prove the following result: Theorem. Every algebraic distributive lattice D with at most ℵ1 compact elements is isomorphic to the ideal lattice of a von Neumann regular ring R. (By earlier results of the author, the ℵ1 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.