1 Correction to: Semigroup Forum (2019) 99:303–316 https://doi.org/10.1007/s00233-018-9967-9
For L a complete lattice, let \({\mathsf {Up}}(L)\) be the complete lattice of all upclosed subsets of L ordered by \(\subseteq ^{op}\). An upclosed subset A is prime if \(a\vee b\in A\) implies \(a\in A\) or \(b\in A\). Let \({\mathsf {PUp}}(L)\) be the complete lattice of all prime members of \({\mathsf {Up}}(L)\).
We apologize for the false statement on page 310 that \({{\mathsf {Up}}(L)\otimes M}\) and \({{\mathsf {PUp}}(L) \otimes M}\) are always order-isomorphic. Indeed, \(M=\varvec{2}\) is the unit of the tensor product. Thus, if that statement were true, this would mean that \({\mathsf {Up}}(L)\) and \({\mathsf {PUp}}(L)\) are order-isomorphic. But this is not always the case as simple counterexamples show. For instance, if L is the four-point diamond, then \({\mathsf {PUp}}(L)\) has five elements, while \({\mathsf {Up}}(L)\) has six elements.
That unfortunate statement played a role at the beginning of the proof of Corollary 2. Since this corollary is perfectly correct, it is the purpose of this note to start the proof of Corollary 2 appropriately. Let f be a tensor of \(M\otimes L\). By complete distributivity of M and Lemma 3, f can uniquely be identified with the map determined by (see (4.4)):
Moreover, since \(A=\mathop {\textstyle \bigcap }\limits _{a\not \in A} L\setminus \downarrow a\), the following hold:
i.e., \(\psi _f\) is uniquely determined by its values at all prime upclosed subsets of the form \(L\setminus \downarrow a\). Hence \(\psi _f\) is completely determined by its restriction to \({\mathsf {PUp}}(L)\). Thus, Lemma 3 implies that every tensor \(f\in M\otimes L\) can uniquely be identified with a map \({\mathsf {PUp}}(L) \xrightarrow {\,\varphi _f\,} M\) determined by (4.5).
The remaining part of the original proof of Corollary 2 remains unchanged.
None of the other results of the paper are affected.
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Communicated by Jimmie D. Lawson.
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Gutiérrez García, J., Höhle, U. & Kubiak, T. Correction to: Some characterizations of the tensor product of complete lattices with applications to quantales. Semigroup Forum 100, 638–639 (2020). https://doi.org/10.1007/s00233-020-10090-w
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DOI: https://doi.org/10.1007/s00233-020-10090-w