Representations of distributive semilattices in ideal lattices of various algebraic structures

Algebra Universalis

Volumn 45, Issue 1, 2001, Pages 71-101

  Goodearl, K R ^a   Wehrung, F ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b UNIVERSITÉ DE CAEN   (France)

Author keywords

Approximately finite dimensional; C* algebra; Compact congruence; Complemented modular lattice; Dimension group; Direct limit; Distributive semilattice; Maximal semilattice quotient; Real rank zero; Von Neumann regular ring

Indexed keywords

EID: 0012585982     PISSN: 00025240     EISSN: None     Source Type: Journal    
DOI: 10.1007/s000120050203     Document Type: Article

References (30)

1. ARA, P., GOODEARL, K. R., O'MEARA, K. C. and PARDO, E., Separative cancellation for projective modules over exchange rings, Israel J. Math. 105 (1998), 105-137.
2. BERGMAN, G. M., Von Neumann regular rings with tailor-made ideal lattices. Unpublished note (26 October 1986).
3. BIRKHOFF, G., "Lattice theory. Corr. repr. of the 1967 3rd ed.", Third Ed., American Math. Soc. Colloq. Publ. 25, Providence, Rhode Island: American Mathematical Society, 1979, vi + 418 pp.
4. BLACKADAR, B., "K-Theory for Operator Algebras", MSRI Publ. 5, Springer-Verlag, New York, Springer-Verlag, 1986, vii + 338 pp.
5. CLIFFORD, A. H. and PRESTON, G. B., "The Algebraic Theory of Semigroups", Mathematical Surveys 7, American Math. Soc., Providence, R.I., vol. 1 (1961), xv + 224 pp., and vol. 2 (1967), xvi + 350 pp.
6. CUNTZ, J., K-theory for certain C*-algebras, Ann. of Math. 113 (1981), 181-197.
7. DAVIDSON, K. R., "C*-Algebras by Example", Fields Inst. Monographs 6, American Math. Soc., Providence, 1996, xiv + 309 pp.
8. DOBBERTIN, H., On Vaught's criterion for isomorphisms of countable Boolean algebras, Algebra Univers. 15 (1982), 95-114.
9. EFFROS, E. G., HANDELMAN, D. E. and SHEN, C. -L., Dimension groups and their affine representations, Amer. J. Math. 102 no. 2 (1980), 385-407.
10. ELLIOTT, G. A., On the classification of inductive limits of sequences of semisimple finite-dimensional algebras, J. Algebra 38 (1976), 29-44.
11. FRYER, K. D. and HALPERIN, I., The von Neumann coordinatization theorem for complemented modular lattices, Acta Sci. Math. (Szeged) 17 (1956), 203-249.
12. GOODEARL, K. R., "Von Neumann Regular Rings", Pitman, London 1979, xvii + 369 pp.; Second Ed. Krieger, Malabar, Fl., 1991, xvi + 412 pp.
13. GOODEARL, K. R., Artinian and nœtherian modules over regular rings, Comm. Algebra 8 (1980), 477-504.
14. GOODEARL, K. R., "Partially Ordered Abelian Groups with Interpolation", Math. Surveys and Monographs 20, Amer. Math. Soc., Providence, 1986, xxii + 336 pp.
15. 0 of unit-regular rings, Canad. J. Math. 38, no. 3 (1986), 633-658.
16. GRÄTZER, G., "Lattice Theory. First Concepts and Distributive Lattices", W. H. Freeman and Co., San Francisco, Calif., 1971, xv + 212 pp.
17. GRÄTZER, G., "General Lattice Theory. Second Edition", Birkhäuser Verlag, Basel, 1998, xix + 663 pp.
18. GRÄTZER, G. and SCHMIDT, E. T., Congruence lattices of lattices, Appendix C in [17].
19. GRILLET, P. A., Directed colimits of free commutative semigroups, J. Pure Appl. Algebra 9 no. 1 (1976), 73-87.
20. HANDELMAN, D. E., Notes on ideal lattices, Unpublished note, 1981.
21. JACOBSON, N., "Structure of Rings", Rev. Ed., Colloq. Publ. 37, Amer. Math. Soc., Providence, 1964, vii + 263 pp.
22. KIM, K. H. and ROUSH, F. W., Regular rings and distributive lattices, Comm. Algebra 8 (1980), 1283-1290.
23. LIN, H., C*-algebras of trivial K-theory and semilattices, Internat. J. Math. 10 (1999), 93-128.
24. MAL'CEV, A. I., "Algebraic Systems", Die Grundlagen der mathematischen Wissenschaften in Einzeldarstellungen, Band 192, Springer-Verlag, Berlin Heidelberg New York, 1973, xii + 317 pp.
25. MURPHY, G. J., "C*-Algebras and Operator Theory", Academic Press, Boston, 1990, x + 286 pp.
26. n, Michigan Math. J. 26 (1979), 3-12.
27. PUDLÁK, P., On congruence lattices of lattices, Algebra Univers. 20 (1985), 96-114.
28. RŮŽIČKA, P., Lattices of two-sided ideals of locally matricial algebras and the Γ-invariant problem, preprint.
29. RŮŽIČKA, P., A distributive semilattice that is not isomorphic to the compact ideal lattice of any dimension group, preprint.
30. SCHMIDT, E. T., Congruence lattices of complemented modular lattices, Algebra Univers. 18 (1984), 386-395.