An Observation on Krull and Derived Dimensions of Some Topological Lattices
نویسندگان
چکیده
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 specL, the set of all prime elements of L. Hence the dimensions on the lattice and on the spectrum coincide.
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