Around the relative center property in orthomodular lattices

Chevalier, G.

doi:10.1090/S0002-9939-1991-1055767-3

Around the relative center property in orthomodular lattices
HTML articles powered by AMS MathViewer

by G. Chevalier PDF

Proc. Amer. Math. Soc. 112 (1991), 935-948 Request permission

Abstract:

The paper deals with the relative center property in orthomodular lattices (OMLs). The property holds in a large class of OMLs, including locally modular OMLs and projection lattices of ${{\text {AW}}^*}$- and ${{\text {W}}^*}$-algebras, and it means that the center of any interval $[0,a]$ is the set $\{ a \wedge c|c\;{\text {central}}\;{\text {in}}\;L\}$. In §l we study the congruence lattice of an OML satisfying the Axiom of Comparability (A.C.) and, in §2, we prove that the central cover of an element can be expressed in many different ways in OMLs satisfying a certain condition (C). For complete OMLs, Axiom (A.C.) and condition (C) are equivalent to the relative center property. In §3, we give a coordinatization theorem for complete OMLs with the relative center property.

References

Donald H. Adams, Equational classes of Foulis semigroups and orthomodular lattices, Proceedings of the University of Houston Lattice Theory Conference (Houston, Tex., 1973) Dept. Math., Univ. Houston, Houston, Tex., 1973, pp. 486–497. MR 0392721
Sterling K. Berberian, Baer *-rings, Die Grundlehren der mathematischen Wissenschaften, Band 195, Springer-Verlag, New York-Berlin, 1972. MR 0429975
S. K. Berberian, Equivalence and perspectivity in Baer rings, J. Algebra 87 (1984), no. 2, 380–388. MR 739941, DOI 10.1016/0021-8693(84)90144-3

Baer rings and Baer

-rings

Anneaux et

-anneaux de Baer

I. Chajda and B. Zelinka, Every finite chain is the tolerance lattice of some lattice, Glas. Mat. Ser. III 20(40) (1985), no. 1, 3–6 (English, with Serbo-Croatian summary). MR 818608
Georges Chevalier, Relations binaires et congruences dans un treillis orthomodulaire, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 19, 785–788 (French, with English summary). MR 711831
Georges Chevalier, Les congruences d’un treillis orthomodulaire de projections, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), no. 15, 731–734 (French, with English summary). MR 772081
Georges Chevalier, Commutators and decompositions of orthomodular lattices, Order 6 (1989), no. 2, 181–194. MR 1031654, DOI 10.1007/BF02034335
D. J. Foulis, Relative inverses in Baer *-semigroups, Michigan Math. J. 10 (1963), 65–84. MR 154939
D. J. Foulis, Semigroups co-ordinatizing orthomodular geometries, Canadian J. Math. 17 (1965), 40–51. MR 204331, DOI 10.4153/CJM-1965-005-4
Richard Greechie and Louis Herman, Commutator-finite orthomodular lattices, Order 1 (1985), no. 3, 277–284. MR 779393, DOI 10.1007/BF00383604
M. F. Janowitz, Separation conditions in relatively complemented lattices, Colloq. Math. 22 (1970), 25–34. MR 280419, DOI 10.4064/cm-22-1-25-34
Gudrun Kalmbach, Orthomodular lattices, London Mathematical Society Monographs, vol. 18, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1983. MR 716496
Irving Kaplansky, Rings of operators, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0244778
Yukiyosi Kawada, Kaneo Higuti, and Yatarô Matusima, Bemerkungen zur vorangehenden Arbeit von Herrn T. Iwamura, Jpn. J. Math. 19 (1944), 73–79 (German). MR 16558, DOI 10.4099/jjm1924.19.1_{7}3
Shûichirô Maeda, On the lattice of projections of a Baer $^*$-ring, J. Sci. Hiroshima Univ. Ser. A 22 (1958), 75–88 (1958). MR 105378
Shûichirô Maeda, On $^{\ast }$-rings satisfying the square root axiom, Proc. Amer. Math. Soc. 52 (1975), 188–190. MR 371941, DOI 10.1090/S0002-9939-1975-0371941-4
Shuichiro Maeda and Samuel S. Holland Jr., Equivalence of projections in Baer $^*$-rings, J. Algebra 39 (1976), no. 1, 150–159. MR 404319, DOI 10.1016/0021-8693(76)90067-3
Yosinao Misonou, On a weakly central operator algebra, Tohoku Math. J. (2) 4 (1952), 194–202. MR 52044, DOI 10.2748/tmj/1178245422
Usa Sasaki, Lattices of projections in $AW^*$-algebras, J. Sci. Hiroshima Univ. Ser. A 19 (1955), 1–30. MR 80641
Dietmar Schweigert, Compatible relations of modular and orthomodular lattices, Proc. Amer. Math. Soc. 81 (1981), no. 3, 462–464. MR 597663, DOI 10.1090/S0002-9939-1981-0597663-5
David M. Topping, Jordan algebras of self-adjoint operators, Mem. Amer. Math. Soc. 53 (1965), 48. MR 190788
Fred B. Wright, A reduction for algebras of finite type, Ann. of Math. (2) 60 (1954), 560—570. MR 65037, DOI 10.2307/1969851
Fred B. Wright, The ideals in a factor, Ann. of Math. (2) 68 (1958), 475–483. MR 100555, DOI 10.2307/1970152