A proof of Weinberg's conjecture on lattice-ordered matrix algebras

Authors:
Jingjing Ma and Piotr J. Wojciechowski

Journal:
Proc. Amer. Math. Soc. **130** (2002), 2845-2851

MSC (2000):
Primary 06F25; Secondary 15A48

DOI:
https://doi.org/10.1090/S0002-9939-02-06408-0

Published electronically:
March 15, 2002

MathSciNet review:
1908906

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a subfield of the field of real numbers and let () be the matrix algebra over . It is shown that if is a lattice-ordered algebra over in which the identity matrix 1 is positive, then is isomorphic to the lattice-ordered algebra with the usual lattice order. In particular, Weinberg's conjecture is true.

**1.**A. Berman and R. J. Plemmons,*Nonnegative Matrices in the Mathematical Sciences*, Academic Press, 1979. MR**82b:15013****2.**G. Birkhoff and R. S. Pierce,*Lattice-ordered rings*, An. Acad. Brasil. Cienc.**28**(1956), 41-69. MR**18:191d****3.**P. Conrad,*Lattice-ordered groups*, Tulane Lecture Notes, Tulane University, 1970.**4.**L. Fuchs,*Partially ordered algebraic systems*, Akademia Kiado, Budapest. MR**30:2090****5.**J. Ma,*Lattice-ordered matrix algebras with the usual lattice order*, J. of Algebra**228**(2000), 406-416. MR**2001d:16066****6.**S. A. Steinberg,*Finitely-valued f-modules*, Pacific J. Math.**40**(1972), 723-737. MR**46:5205****7.**S. A. Steinberg,*On the scarcity of lattice-ordered matrix algebras II*, Proc. Amer. Math. Soc.**128**(2000), no. 6, 1605-1612. MR**2000j:06011****8.**E. C. Weinberg,*On the scarcity of lattice-ordered matrix rings*, Pacific J. Math.**19**(1966), 561-571. MR**34:2635**

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Additional Information

**Jingjing Ma**

Affiliation:
Department of Mathematical Sciences, University of Houston-Clear Lake, 2700 Bay Area Boulevard, Houston, Texas 77058

Email:
ma@cl.uh.edu

**Piotr J. Wojciechowski**

Affiliation:
Department of Mathematical Sciences, The University of Texas at El Paso, El Paso, Texas 79968

Email:
piotr@math.utep.edu

DOI:
https://doi.org/10.1090/S0002-9939-02-06408-0

Keywords:
Lattice-ordered algebra,
matrix algebra

Received by editor(s):
March 20, 2001

Received by editor(s) in revised form:
May 16, 2001

Published electronically:
March 15, 2002

Additional Notes:
The results in this paper were presented at the conference “Lattice-ordered groups and f-rings" at the University of Florida, March 2001.

Dedicated:
Dedicated to Professor Melvin Henriksen on his 75th birthday

Communicated by:
Lance W. Small

Article copyright:
© Copyright 2002
American Mathematical Society