Arens regularity of lattice-ordered rings

Karim Boulabiar[1]; Jamel Jabeur[2]

  • [1] Département du Cycle Agrégatif IPEST, Université du 7 Novembre à Carthage BP 51, 2070-La Marsa, Tunisia
  • [2] Département du Cycle Préparatoire IPEST, Université du 7 Novembre à Carthage BP 51, 2070-La Marsa, Tunisia

Annales de la faculté des sciences de Toulouse Mathématiques (2010)

  • Volume: 19, Issue: S1, page 25-36
  • ISSN: 0240-2963

Abstract

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This work discusses the problem of Arens regularity of a lattice-ordered ring. In this prospect, a counterexample is furnished to show that without extra conditions, a lattice-ordered ring need not be Arens regular. However, as shown in this paper, it turns out that any f -ring in the sense of Birkhoff and Pierce is Arens regular. This result is then used and extended to the more general setting of almost f -rings introduced again by Birkhoff.

How to cite

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Boulabiar, Karim, and Jabeur, Jamel. "Arens regularity of lattice-ordered rings." Annales de la faculté des sciences de Toulouse Mathématiques 19.S1 (2010): 25-36. <http://eudml.org/doc/115900>.

@article{Boulabiar2010,
abstract = {This work discusses the problem of Arens regularity of a lattice-ordered ring. In this prospect, a counterexample is furnished to show that without extra conditions, a lattice-ordered ring need not be Arens regular. However, as shown in this paper, it turns out that any $f$-ring in the sense of Birkhoff and Pierce is Arens regular. This result is then used and extended to the more general setting of almost $f$-rings introduced again by Birkhoff.},
affiliation = {Département du Cycle Agrégatif IPEST, Université du 7 Novembre à Carthage BP 51, 2070-La Marsa, Tunisia; Département du Cycle Préparatoire IPEST, Université du 7 Novembre à Carthage BP 51, 2070-La Marsa, Tunisia},
author = {Boulabiar, Karim, Jabeur, Jamel},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {almost -ring; Arens regular ring; unital -ring; lattice-ordered ring},
language = {eng},
month = {4},
number = {S1},
pages = {25-36},
publisher = {Université Paul Sabatier, Toulouse},
title = {Arens regularity of lattice-ordered rings},
url = {http://eudml.org/doc/115900},
volume = {19},
year = {2010},
}

TY - JOUR
AU - Boulabiar, Karim
AU - Jabeur, Jamel
TI - Arens regularity of lattice-ordered rings
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2010/4//
PB - Université Paul Sabatier, Toulouse
VL - 19
IS - S1
SP - 25
EP - 36
AB - This work discusses the problem of Arens regularity of a lattice-ordered ring. In this prospect, a counterexample is furnished to show that without extra conditions, a lattice-ordered ring need not be Arens regular. However, as shown in this paper, it turns out that any $f$-ring in the sense of Birkhoff and Pierce is Arens regular. This result is then used and extended to the more general setting of almost $f$-rings introduced again by Birkhoff.
LA - eng
KW - almost -ring; Arens regular ring; unital -ring; lattice-ordered ring
UR - http://eudml.org/doc/115900
ER -

References

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  1. C. D. Aliprantis, and O. Burkinshaw, Positive operators, Academic Press, Orlando, 1985 . Zbl0608.47039MR809372
  2. R. Arens, The adjoint of bilinear operation, Proc. Amer. Math. Soc., 2 ( 1951 ), 839 - 848 . Zbl0044.32601MR45941
  3. A. Bigard, K. Keimel, and S. Wolfenstein, Groupes et Anneaux Réticulés, Lecture Notes Math. 608 , Springer Verlag, Berlin-Heidelberg-New York, 1977 . Zbl0384.06022MR552653
  4. G. Birkhoff, Lattice Theory, 3rd. edition Am. Math. Soc. Colloq. Publ. No. 25 , Providence, Rhode Island, 1967 . MR227053
  5. G. Birkhoff and R. S. Pierce, Lattice-ordered rings, An. Acad. Brasil. Ciènc. 28 ( 1956 ), 41 - 69 . Zbl0070.26602MR80099
  6. K. Boulabiar, Representation theorems for d -multiplications on Archimedean unital f -rings, Comm. Algebra, 32 ( 2004 ), 3955 - 3967 . Zbl1060.06023MR2097440
  7. K. Boulabiar and J. Jabeur, Arens regularity of lattice-ordered rings, Ann. Fac. Sci. Toulouse, Math., To appear. Zbl1210.06010
  8. G. Buskes and R. Page, A positive note on a couterexample by Arens, Quaest. Math., 28 ( 2005 ), 117 - 121 . Zbl1080.47003MR2136186
  9. P. Conrad, The additive group of an f -ring, Canad. J. Math., 26 ( 1974 ), 1157 - 1168 . Zbl0293.06019MR354487
  10. C. B. Huijsmans and B. de Pagter, The order bidual of lattice ordered algebras, J. Funct. Anal., 59 ( 1984 ), 41 - 64 . Zbl0549.46006MR763776

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