An observation on Krull and derived dimensions of some topological lattices

M. Rostami; Ilda I. Rodrigues

Archivum Mathematicum (2011)

  • Volume: 047, Issue: 4, page 329-334
  • ISSN: 0044-8753

Abstract

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Let ( L , ≤ ) , be an algebraic lattice. It is well-known that ( L , ≤ ) with its topological structure is topologically scattered if and only if ( L , ≤ ) is ordered scattered with respect to its algebraic structure. In this note we prove that, if L is a distributive algebraic lattice in which every element is the infimum of finitely many primes, then L has Krull-dimension if and only if L has derived dimension. We also prove the same result for error L , the set of all prime elements of L . Hence the dimensions on the lattice and on the spectrum coincide.

How to cite

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Rostami, M., and Rodrigues, Ilda I.. "An observation on Krull and derived dimensions of some topological lattices." Archivum Mathematicum 047.4 (2011): 329-334. <http://eudml.org/doc/246295>.

@article{Rostami2011,
abstract = {Let $(L, \le )$, be an algebraic lattice. It is well-known that $(L, \le )$ with its topological structure is topologically scattered if and only if $(L, \le )$ is ordered scattered with respect to its algebraic structure. In this note we prove that, if $L$ is a distributive algebraic lattice in which every element is the infimum of finitely many primes, then $L$ has Krull-dimension if and only if $L$ has derived dimension. We also prove the same result for $\operatorname\{\it spec\} L$, the set of all prime elements of $L$. Hence the dimensions on the lattice and on the spectrum coincide.},
author = {Rostami, M., Rodrigues, Ilda I.},
journal = {Archivum Mathematicum},
keywords = {Krull dimension; derived dimension; inductive dimension; scattered spaces and algebraic lattices; Krull dimension; derived dimension; inductive dimension; scattered space; algebraic lattice; topological lattice},
language = {eng},
number = {4},
pages = {329-334},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {An observation on Krull and derived dimensions of some topological lattices},
url = {http://eudml.org/doc/246295},
volume = {047},
year = {2011},
}

TY - JOUR
AU - Rostami, M.
AU - Rodrigues, Ilda I.
TI - An observation on Krull and derived dimensions of some topological lattices
JO - Archivum Mathematicum
PY - 2011
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 047
IS - 4
SP - 329
EP - 334
AB - Let $(L, \le )$, be an algebraic lattice. It is well-known that $(L, \le )$ with its topological structure is topologically scattered if and only if $(L, \le )$ is ordered scattered with respect to its algebraic structure. In this note we prove that, if $L$ is a distributive algebraic lattice in which every element is the infimum of finitely many primes, then $L$ has Krull-dimension if and only if $L$ has derived dimension. We also prove the same result for $\operatorname{\it spec} L$, the set of all prime elements of $L$. Hence the dimensions on the lattice and on the spectrum coincide.
LA - eng
KW - Krull dimension; derived dimension; inductive dimension; scattered spaces and algebraic lattices; Krull dimension; derived dimension; inductive dimension; scattered space; algebraic lattice; topological lattice
UR - http://eudml.org/doc/246295
ER -

References

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  1. Birkhoff, G., Lattice Theory, New York, Providence AMS, 1940. (1940) Zbl0063.00402MR0001959
  2. Erné, M., Gehrke, M., Pultr, A., 10.1007/s10485-006-9054-3, Appl. Categ. Structures 15 (2007), 163–184. (2007) Zbl1122.06015MR2306544DOI10.1007/s10485-006-9054-3
  3. Gierz, G., Keimel, K., 10.1007/BFb0089905, Lecture Notes in Math. 871 (1971), 97–124. (1971) DOI10.1007/BFb0089905
  4. Gierz, G. et al.,, A Compendium of Continuous Lattices, Springer–Verlag, New York, 1980. (1980) Zbl0452.06001MR0614752
  5. Hausdorff, F., 10.1007/BF01451165, Math. Ann. 65 (4) (1908), 435–505. (1908) MR1511478DOI10.1007/BF01451165
  6. Johnstone, P., Stone Spaces, Cambridge Stud. Adv. Math., 3, Cambridge University Press, 1986. (1986) Zbl0586.54001MR0861951
  7. Karamzadeh, O. A. S., On the classical Krull dimension of rings, Fund. Math. 117 (2) (1983), 103–108. (1983) Zbl0542.16022MR0719833
  8. Mislove, M., When are order scattered and topologically scattered the same?, Orders: Description and Roles (Pouzet, M., Richard, D., eds.), North–Holland Math. Stud., 1984, pp. 61–80. (1984) Zbl0553.06007MR0779845
  9. Mislove, M., Order–scattered distributive continuous lattices are topologically scattered, Houston J. Math. 11 (4) (1985), 559–573. (1985) Zbl0595.06011MR0837993
  10. Mislove, M., 10.1016/S0166-8641(97)00222-8, Topology Appl. 89 (1–2) (1998), 3–59. (1998) MR1641441DOI10.1016/S0166-8641(97)00222-8
  11. Năstăsescu, C., Van Oystaeyen, F., Dimensions of ring theory, Mathematics and its Applications, 36, D. Reidel Publishing Company, Dordrecht, 1987. (1987) MR0894033
  12. Niefield, S. B., Rosenthal, K. I., 10.1016/0166-8641(87)90046-0, Topology Appl. 26 (3) (1987), 263–269. (1987) Zbl0621.06007MR0904472DOI10.1016/0166-8641(87)90046-0
  13. Puczylowski, E. R., Gabriel and Krull dimensions of modules over rings graded by finite groups, Proc. Amer. Math. Soc. 105 (4) (1985), 17–224. (1985) MR0973835
  14. Simmons, H., 10.1017/S0013091500015868, Proc. Edinb. Math. Soc. 21 (1987), 41–48. (1987) MR0493959DOI10.1017/S0013091500015868

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