| Abstract:
In \(S_0\), the category of all join semilattices with 0, the tensor product has
all the usual properties as in the category of abelian groups. Since join
semilattices are locally finite and finite 0-semilattices are lattices, so
the tensor product of two finite join semilattices is finite and a lattice.
This result does not carry over to the infinite case since, as shown by G.
Gratzer and F. Wehrung, the tensor product of \(M_3\), the 5-element lattice
with 3 atoms, and \(F_3\), the free lattice on 3 generators, is not a lattice.
They then define a related construction, called the lattice tensor product,
and give a very computational proof that it always yields a lattice. This
paper gives an easier, more conceptual, proof using the concept of an upper
bounded join homomorphism.
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