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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