Order-topological complete orthomodular lattices
The authors prove the equivalence of the following statements for a complete orthomodular lattice \(L\): (i) \(L\) is order-topological; (ii) \(L\) is (Scott-)continuous; (iii) \(L\) is algebraic; (iv) \(L\) is compactly atomistic; (v) \(L\) is in the order topology a totally order-disconnected topological lattice. A lattice \(L\) (element \(a\in L\)) is hypercompact, if its hypercompact elements are join-dense (the complement of \([a)\) is a finite union of principal ideals). For an orthomodular lattice \(L\), the following conditions are equivalent: (i) \(L\) is hypercompactly generated; (ii) \(L\) is atomic and for every atom \(a\in L\), \(N_ a= \{b\in A| a\not\leq b'\}\) is finite; (iii) \(L\) is meet-continuous, atomic with a \(T_ 2\)- interval topology; (iv) the MacNeille completion of \(L\) is a compact \(T_ 1\) (\(T_ 2\), totally disconnected) OML.
scientific article; zbMATH DE number 735682
scientific article