An observation on Krull and derived dimensions of some topological lattices

M. Rostami and Ilda I. Rodrigues

Address: Departamento de Matemática, Universidade da Beira Interior, 6200 Covilhã, Portugal


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.

AMSclassification: primary 06B30; secondary 16U20, 54G12, 06-XX, 54C25.

Keywords: Krull dimension, derived dimension, inductive dimension, scattered spaces and algebraic lattices.