On a multiplication decomposition theorem in a Dedekind -complete partially ordered linear algebra

Author:
Taen Yu Dai

Journal:
Proc. Amer. Math. Soc. **44** (1974), 12-16

MSC:
Primary 06A70

DOI:
https://doi.org/10.1090/S0002-9939-1974-0335393-1

MathSciNet review:
0335393

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Abstract: Suppose a Dedekind -complete partially ordered linear algebra (dsc-pola) satisfies a certain multiplication decomposition property (see definition below), then we show that this partially ordered linear algebra actually has the same structure of a special class of real matrix algebras, consisting of elements that can be decomposed as diagonal part plus nilpotent part , such that .

**[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]**T. Y. Dai,*On some specially classes of partially ordered linear algebras*, J. Math. Anal. Appl.**40**(1972), 649-682. MR**0316342 (47:4890)****[3]**R. E. DeMarr,*On partially ordering operator algebras*, Canad. J. Math.**19**(1967), 636-643. MR**35**#3450. MR**0212579 (35:3450)****[4]**-,*A class of partially ordered linear algebras*, Proc. Amer. Math. Soc.**39**(1973), 255-260. MR**0313161 (47:1716)**

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DOI:
https://doi.org/10.1090/S0002-9939-1974-0335393-1

Keywords:
Dedekind -complete partially ordered linear algebra,
nilpotent,
multiplication decomposition property,
matrix inequalities

Article copyright:
© Copyright 1974
American Mathematical Society