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