Representation of algebraic distributive lattices with ℵ1 compact elements as ideal lattices of regular rings.

Wehrung, Friedrich. "Representation of algebraic distributive lattices with ℵ1 compact elements as ideal lattices of regular rings.." Publicacions Matemàtiques 44.2 (2000): 419-435. <http://eudml.org/doc/41405>.

@article{Wehrung2000,
abstract = {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. Tuma, 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.},
author = {Wehrung, Friedrich},
journal = {Publicacions Matemàtiques},
keywords = {Retículos; Anillos reticulados; Anillos regulares; von Neumann regular rings; ideal lattices; algebraic lattices; congruence lattices; modular lattices; compact elements; distributive lattices; semilattice amalgamations},
language = {eng},
number = {2},
pages = {419-435},
title = {Representation of algebraic distributive lattices with ℵ1 compact elements as ideal lattices of regular rings.},
url = {http://eudml.org/doc/41405},
volume = {44},
year = {2000},
}

TY - JOUR
AU - Wehrung, Friedrich
TI - Representation of algebraic distributive lattices with ℵ1 compact elements as ideal lattices of regular rings.
JO - Publicacions Matemàtiques
PY - 2000
VL - 44
IS - 2
SP - 419
EP - 435
AB - 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. Tuma, 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.
LA - eng
KW - Retículos; Anillos reticulados; Anillos regulares; von Neumann regular rings; ideal lattices; algebraic lattices; congruence lattices; modular lattices; compact elements; distributive lattices; semilattice amalgamations
UR - http://eudml.org/doc/41405
ER -