Tensor products of semilattices with zero, revisited

George Grätzer; Friedrich Wehrung

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arXiv · math/0501436

Tensor products of semilattices with zero, revisited

George Grätzer·Friedrich Wehrung

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Abstract

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.

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George Grätzer, Friedrich Wehrung. 2005-01-25. Tensor products of semilattices with zero, revisited. https://arxiv.org/abs/math/0501436

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