Geodesic
An observation on Krull and derived dimensions of some topological lattices
Archivum mathematicum, Tome 47 (2011) no. 4, pp. 329-334
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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.
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.
Classification : 06-xx, 06B30, 16U20, 54C25, 54G12
Keywords: Krull dimension; derived dimension; inductive dimension; scattered spaces and algebraic lattices 
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Rostami, M.; Rodrigues, Ilda I. An observation on Krull and derived dimensions of some topological lattices. Archivum mathematicum, Tome 47 (2011) no. 4, pp. 329-334. http://geodesic.mathdoc.fr/item/ARM_2011_47_4_a7/

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