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Note that all the semilattices which we deal with turn out to be lattices, however the poset maps to be considered will preserve meets, but usually not joins, thus we really work in the category of meet-semilattices.
Here is Auslander's first main assertion: [ → Y 〉 = ⋃ C C [ → Y 〉, (1 where C runs through all isomorphism classes of Λ -modules; or, in the formulation of Auslander: any map in mod Λ is right determined by some module C. Note that the inclusion of the lattice C [ → Y 〉 into the lattice [ → Y 〉 preserves meets, but usually not joins.
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