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Almost f-algebras and d-algebras

Published online by Cambridge University Press:  24 October 2008

S. J. Bernau  and
C. B. Huijsmans
S. J. Bernau
Affiliation:
Department of Mathematical Sciences, University of Texas at El Paso, El Paso, Texas 79968-0514, U.S.A.
C. B. Huijsmans
Affiliation:
Department of Mathematics, Rijksuniversiteit Leiden, P.O. Box 9512, 2300 RA Leiden, The Netherlands
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Extract

In this paper we study the classes of almost f-algebras and d-algebras. Apart from a survey of known properties we present some ‘new’ properties of almost f-algebras and d-algebras and we consider their connection with f-algebras. To be more precise, we show, among other things, in an elementary intrinsic manner, that every Archimedean almost f-algebra is commutative. This result is a considerable improvement upon the fact that every Archimedean f-algebra is commutative. Furthermore, we give a description of the set of nilpotent elements in both an Archimedean d-algebra and an Archimedean almost f-algebra.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 107 , Issue 2 , March 1990 , pp. 287 - 308
Copyright
Copyright © Cambridge Philosophical Society 1990

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References

REFERENCES

[1]Basly, M. and Triki, A.. FF-Algèbres Archimédiennes réticulées. (Preprint, 1988.)Google Scholar
[2]Basly, M., Huijsmans, C. B., de Pagter, B. and Triki, A.. On unital Archimedean lattice-ordered algebras. J. Fac. Sci. Tunis (to appear).Google Scholar
[3]Bernau, S. J.. On semi-normal lattice rings. Proc. Cambridge Philos. Soc. 61 (1965), 613–616.CrossRefGoogle Scholar
[4]Bigard, A., Keimel, K. and Wolfenstein, S.. Groupes et Anneaux Réticulés. Lecture Notes in Math. vol. 608 (Springer-Verlag, 1977).CrossRefGoogle Scholar
[5]Birkhoff, G.. Lattice Theory. 3rd Edition. Amer. Math. Soc. Colloq. Publ. no. 25 (American Mathematical Society, 1967).Google Scholar
[6]Birkhoff, G. and Pierce, R. S.. Lattice-ordered rings. An. Acad. Brasil. Ciènc. 28 (1956), 41–69.Google Scholar
[7]Diem, J. E.. A radical for lattice-ordered rings. Pacific J. Math. 25 (1968), 71–82.CrossRefGoogle Scholar
[8]Huijsmans, C. B. and de Pagter, B.. Ideal theory in f-algebras. Trans. Amer. math. Soc. 269 (1982), 225–245.Google Scholar
[9]Huijsmans, C. B. and de Pagter, B.. Averaging operators and positive contractive projections. J. Math. Anal. Appl. 113 (1986), 163–184.CrossRefGoogle Scholar
[10]Kudláček, V.. O některých typech 1-okruhu (on some types of l-rings). Sborní Vysokého Učeni Techn. υ Brně 1–2 (1962), 179–181.Google Scholar
[11]Luxemburg, W. A. J.. Some Aspects of the Theory of Riesz Spaces. Univ. Arkansas Lecture Notes Math. vol. 4 (Fayetteville, 1979).Google Scholar
[12]Luxemburg, W. A. J. and Zaanen, A. C.. Riesz Spaces, vol. 1 (North-Holland, 1971).Google Scholar
[13]Martignon, L. F.. Banachverbandsalgebren. Ph.D. thesis, Universität Tübingen (1978).Google Scholar
[14]de Pagter, B.. F-Algebras and orthomorphisms. Ph.D. thesis, University of Leiden (1981).Google Scholar
[15]Quint, C.. Zur Darstelling von Banachverbandsalgebren. Ph.D. thesis, Universität Darmstadt (1984).Google Scholar
[16]Scheffold, E.. FF-Banachverbandsalgebren. Math. Z. 177 (1981), 193–205.CrossRefGoogle Scholar
[17]Scorza, G.. Le algebre del 4° ordine. Mem. Reale Accad. Scienze Fis. Mat. Napoli (2) 20 (1935), n. 14;Google Scholar
Opere Scelte III (Edizioni Cremonese, 1962), 259–361.Google Scholar
[18]Steinberg, S. A.. On lattice-ordered rings in which the square of every element is positive. J. Austral. Math. Soc. Ser. A 22 (1976), 362–370.CrossRefGoogle Scholar
[19]Steinberg, S. A.. Radical theory in lattice-ordered rings. In Gruppi e Anelli Ordinati, Symposia Math. XXI (Academic Press, 1977), pp. 379–400.Google Scholar
[20]Viswanathan, T. M.. Ordered modules of fractions. J. Reine Angew. Math. 235 (1969), 78–107.Google Scholar
[21]Zaanen, A. C.. Riesz Spaces, vol. 2 (North-Holland, 1983).Google Scholar