A property for inverses in a partially ordered linear algebra

Authors:
Taen Yu Dai and Ralph DeMarr

Journal:
Trans. Amer. Math. Soc. **215** (1976), 285-292

MSC:
Primary 06A70

DOI:
https://doi.org/10.1090/S0002-9947-1976-0382116-2

MathSciNet review:
0382116

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Abstract: We consider a Dedekind -complete partially ordered linear algebra *A* which has the following property: if and , then , where . This property is used to show that *A* must be commutative. We also show that *A* is the direct sum of two algebras, each of which behaves like an algebra of real-valued functions.

**[1]**G. Birkhoff,*Lattice theory*, 3rd ed., Amer. Math. Soc. Colloq. Publ., vol. 25, Amer. Math. Soc., Providence, R. I., 1967. MR**37**#2638. MR**0227053 (37:2638)****[2]**Taen-yu Dai,*On some special classes of partially ordered linear algebras*, J. Math. Anal. Appl.**40**(1972), 649-682. MR**0316342 (47:4890)****[3]**Ralph DeMarr,*A class of partially ordered linear algebras*, Proc. Amer. Math. Soc.**39**(1973), 255-260. MR**47**#1716. MR**0313161 (47:1716)****[4]**D. W. Duboic,*A note on David Harrison's theory of preprimes*, Pacific J. Math.**21**(1967), 15-19. MR**35**#103. MR**0209200 (35:103)****[5]**-,*Second note on David Harrison's theory of preprimes*, Pacific J. Math.**24**(1968), 57-68. MR**36**#5049. MR**0221997 (36:5049)****[6]**R. E. Jamison,*Functional representation of algebraic intervals*, Pacific J. Math. (to appear). MR**0365145 (51:1398)****[7]**R. V. Kadison,*A representation theory for commutative topological algebra*, Mem. Amer. Math. Soc. No. 7 (1951). MR**13**, 360. MR**0044040 (13:360b)****[8]**H. H. Schaefer,*Topological vector spaces*, Macmillan, New York, 1966. MR**33**#1689. MR**0193469 (33:1689)****[9]**B. Z. Vulih,*Introduction to the theory of partially ordered spaces*, Fizmatgiz, Moscow, 1961; English transl., Wolters-Noordhoff, Groningen, 1967. MR**24**#A3494;**37**#121. MR**0224522 (37:121)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1976-0382116-2

Keywords:
Dedekind -complete partially ordered linear algebra,
algebra of real-valued functions,
*f*-ring,
matrix inequalities

Article copyright:
© Copyright 1976
American Mathematical Society