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

Publicacions Matemàtiques (2000)

- Volume: 44, Issue: 2, page 419-435
- ISSN: 0214-1493

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topWehrung, 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 -

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