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Topological difference posets. (English) Zbl 1018.06501
Summary: In this paper, some topological properties of difference posets (D-posets) are studied. The basic notions are: a topological D-poset (a D-poset with a topology guaranteeing the continuity of the difference operation), a topological lattice D-poset (a lattice D-poset with a topology guaranteeing the continuity of the difference operation and lattice operations), a uniform D-poset and uniform lattice D-poset (if the topologies are uniform and the operations are uniformly continuous). The main result is the theorem asserting that the topological completion of a uniform Hausdorff lattice D-poset in which all monotone nets are Cauchy is also a uniform Hausdorff lattice D-poset, which is a complete lattice. This is the generalization of a known result for orthomodular lattices.

MSC:
06B30 Topological lattices
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
03G12 Quantum logic
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