Some embedding theorems for lattices
In this thesis we give a general definition of a geometry on a set S and consider the lattices of the subspaces of these geometries. First, we show that all such geometries on a fixed set S form a lattice and we investigate its properties. Secondly, we show that the lattice of all geometries on a...
| Summary: | In this thesis we give a general definition of a geometry on a set S and consider the lattices of the subspaces of these geometries.
First, we show that all such geometries on a fixed set S form a lattice and we investigate its properties.
Secondly, we show that the lattice of all geometries on a fixed set S is isomorphic to the lattice of subspaces of some geometry and we characterize all such geometries.
Finally, we show that every finite lattice can be embedded in the lattice of all geometries on some finite set S. This reduces the unsolved problem of embedding every finite lattice into a finite partition lattice to the problem of embedding every finite lattice of geometries into a finite partition lattice. |
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