Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On the scarcity
of lattice-ordered matrix algebras II


Author: Stuart A. Steinberg
Journal: Proc. Amer. Math. Soc. 128 (2000), 1605-1612
MSC (1991): Primary 06F25; Secondary 15A48
DOI: https://doi.org/10.1090/S0002-9939-99-05171-0
Published electronically: September 23, 1999
MathSciNet review: 1641109
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We correct and complete Weinberg's classification of the lattice-orders of the matrix ring ${\Bbb Q}_2$ and show that this classification holds for the matrix algebra $F_2$ where $F$ is any totally ordered field. In particular, the lattice-order of $F_2$ obtained by stipulating that a matrix is positive precisely when each of its entries is positive is, up to isomorphism, the only lattice-order of $F_2$ with $1>0$. It is also shown, assuming a certain maximum condition, that $(F^+)_n$ is essentially the only lattice-order of the algebra $F_n$ in which the identity element is positive.


References [Enhancements On Off] (What's this?)

  • 1. A. A. Albert, On ordered algebras, Bull. Am. Math. Soc. 46 (1940), 521-522.MR 1:328e
  • 2. G. Birkhoff and R. S. Pierce, Lattice-ordered rings, An. Acad. Brasil 28 (1956), 41-69.MR 18:191d
  • 3. P. Conrad, The lattice of all convex $\ell$-subgroups of a lattice-ordered group, Czechoslovak Math. J. 15 (1965), 101-123. MR 30:3926
  • 4. P. Conrad, Lattice-ordered groups, Tulane Lecture Notes, Tulane University, 1970.
  • 5. L. Fuchs, Partially ordered algebraic systems, Akademia Kiadó, Budapest, 1963. MR 30:2090
  • 6. M. Henriksen and J.R. Isbell, Lattice-ordered rings and function rings, Pacific J. Math. 12 (1962), 533-565. MR 27:3670
  • 7. D. G. Johnson, A structure theory for a class of lattice-ordered rings, Acta. Math. 104 (1960), 163-215. MR 23:A2447
  • 8. S. A. Steinberg, Finitely-valued f-modules, Pacific J. Math. 40 (1972), 723-737. MR 46:5205
  • 9. S. A. Steinberg, Unital $\ell$-prime lattice-ordered rings with polynomial constraints are domains, Trans. Amer. Math. Soc. 276 (1983), 145-164. MR 84d:16050
  • 10. S. A. Steinberg, Central f-elements in lattice-ordered algebras, Ordered Algebraic Structures, Eds. J. Martinez and C. Holland, Kluwer (1993), 203-223.MR 96d:06023
  • 11. M. V. Tamhankar, On algebraic extensions of subrings in an ordered ring, Algebra Universalis 14 (1982), 25-35. MR 83f:06029
  • 12. E. C. Weinberg, On the scarcity of lattice-ordered matrix rings, Pacific J. Math. 19 (1966), 561-571. MR 34:2635

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 06F25, 15A48

Retrieve articles in all journals with MSC (1991): 06F25, 15A48


Additional Information

Stuart A. Steinberg
Affiliation: Department of Mathematics, The University of Toledo, Toledo, Ohio 43606-3390
Email: ssteinb@uoft02.utoledo.edu

DOI: https://doi.org/10.1090/S0002-9939-99-05171-0
Keywords: Lattice-ordered algebra, matrix algebra
Received by editor(s): March 27, 1998
Received by editor(s) in revised form: July 17, 1998
Published electronically: September 23, 1999
Communicated by: Ken Goodearl
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society