Topological Duality for Distributive Lattices: Theory and Applications

This page accompanies the book Topological Duality for Distributive Lattices: Theory and Applications, by Mai Gehrke and Sam van Gool (2024), available for purchase from Cambridge University Press via the above link.

Errata

If you find mistakes, or have questions about the book's contents, please write to us at vangool@irif.fr and mai.gehrke@unice.fr.

• Acknowledgments: the second letter of Mirna Džamonja's last name was unfortunately misprinted.
• p. 5, Example 1.12: replace "any subset has an infimum, which is in fact a minimum" by "any non-empty subset has an infimum, which is in fact a minimum".
• p. 8, Exercise 1.18(a): insert a comma after "order preserving".
• p. 9, item (e), the equation should be $\top \wedge a = a$.
• p. 11, l. 10: insert "are" between $a \wedge b$ and "in".
• p. 26: the items of Exercise 1.3.8 should be labeled (a), (b), (c), (d).
• p. 35, Exercise 2.1.6 (d): there should be an intersection instead of a union. In the same exercise (f), there should be an intersection before D.
• p. 45, line 3: there is a typo in the word "coherent".
• p. 49, Exercise 2.3.5: replace "an arbitrary Cartesian product" by "arbitrary Cartesian products".
• p. 63, line below Proposition 3.15: "Proposition 3.5" should be "Proposition 3.15".
• p. 118, line above Definition 4.44: "Theorem 5.41" should be "Theorem 4.41".
• p. 276, two lines after the first displayed equation on this page, $\delta_x$ should be $\delta_w$.
• p. 248, Theorem 7.38: in this statement, the topology tau is of course also necessarily compact. (This is part of the definition of a coherent space.)
• p. 294, Remark 8.29: "see Exercise 8.2.11" should be "see Exercise 8.2.12".
• p. 297, last equation display: the lower case p should be upper case.
• p. 301, 4 lines before the end of the proof of Proposition 8.39, "because if $\theta \subseteq \theta'$" should read "because if $\theta’ \subseteq \theta$".
• p. 317, Remark 8.56: The reference (Pouzet, 2023) is missing from the bibliography; it should point to the slides available here: http://math.univ-lyon1.fr/~pouzet/wqobqo/Exposes/Pouzet.pdf.
• p. 319, line after Corollary 8.62, the first sentence should read: "It now remains to show that $J_triv \subseteq PT$.

Hints

• p. 17, Exercise 1.2.10 is difficult. See, for example, Burris and Sankappanavar 2000, I.3 for a detailed proof.
• p. 18, Exercise 1.2.16, here is a hint: Show first that $x (\theta_1 \vee \theta_2) y$ if, and only if, there exists a chain of elements $x \wedge y = z_0 \leq \cdots \leq z_n = x \vee y$ in $L$ such that, for any $0 \leq k < n$, we have $z_k (\theta_1 \cup \theta_2) z_{k+1}$.
• p. 25, Exercise 1.3.4(d): instead of "an example as in (c)", look for a frame which has enough meet irreducibles but not enough join irreducibles. We do not know if a frame with enough join irreducibles but not enough meet irreducibles exists.

Acknowledgments

We thank the following people for contributing to the above errata: Jérémie Marquès, Artur Szafarczyk. If you find mistakes please let us know at vangool@irif.fr and mai.gehrke@unice.fr.

Preprint version (old)

If you want to "try before you buy", an older, pre-print version is available on ArXiv.
(Caution: the ArXiv version does not reflect the latest corrections.)