Address: Departamento de Matemática, Universidade da Beira Interior, 6200 Covilhã, Portugal
E-mail:
rostami@ubi.pt
ilda@ubi.pt
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.