Mathematics – General Mathematics
Scientific paper
2005-01-25
Journal of Pure and Applied Algebra 147 (2000) 273--301
Mathematics
General Mathematics
Scientific paper
Let A and B be lattices with zero. The classical tensor product, $A\otimes B$, of A and B as join-semilattices with zero is a join-semilattice with zero; it is, in general, not a lattice. We define a very natural condition: $A \otimes B$ is capped (that is, every element is a finite union of pure tensors) under which the tensor product is always a lattice. Let Conc L denote the join-semilattice with zero of compact congruences of a lattice L. Our main result is that the following isomorphism holds for any capped tensor product: $Conc A\otimes Conc B \cong Conc(A \otimes B)$. This generalizes from finite lattices to arbitrary lattices the main result of a joint paper by the first author, H. Lakser, and R. W. Quackenbush.
Grätzer George
Wehrung Friedrich
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