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

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Abstract | References | Similar Articles | Additional Information

Abstract: We correct and complete Weinberg's classification of the lattice-orders of the matrix ring and show that this classification holds for the matrix algebra where is any totally ordered field. In particular, the lattice-order of 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 with . It is also shown, assuming a certain maximum condition, that is essentially the only lattice-order of the algebra in which the identity element is positive.

**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 -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 -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**

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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