Abstract
Urquhart works in several areas of logic where he has proved important results. Our paper outlines his topological lattice representation and attempts to relate it to other lattice representations. We show that there are different ways to generalize Priestley’s representation of distributive lattices—Urquhart’s being one of them, which tries to keep prime filters (or their generalizations) in the representation. Along the way, we also mention how semi-lattices and lattices figured into Urquhart’s work.
Second Reader
I. Düntsch
Fujian Normal University & Brock University
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Notes
- 1.
The “LLL” was founded by Robert K. Meyer in 1969. The LLL’s manifesto, group pictures of some of the members, and more may be found at the url (as of June 2020): aal.ltumathstats.com/curios/logicians-liberation-league.
- 2.
We will call it Lepidopterary to attract scientifically minded visitors. :-)
- 3.
In other words, every group is a subgroup of the automorphism group on some set; “automorphism” is the taxonomical name for a permutation in the scheme of morphisms.
- 4.
Sometimes the Lindenbaum algebra is called Lindenbaum–Tarski algebra.
- 5.
We use the notation \(\mathscr {C}\) as in Bimbó and Dunn (2008), that is, \(C\in \mathscr {C}\) iff C is a cone (or an upset, or an increasing set—to use other terms). Then, \(\mathscr {C}_1\) and \(\mathscr {C}_2\) are the sets of cones with respect to \(\sqsubseteq _1\) and \(\sqsubseteq _2\), respectively. We may omit parentheses—for readability—from r(V) and l(V).
- 6.
A Stone space for a Boolean algebra is a compact totally disconnected topology. But for a distributive lattice, Stone gave a more complicated characterization. Namely, the topology is \(T_0\) with a basis comprising relatively bicompact sets with a further property linking intersections of basic sets with a closed set.
- 7.
At least, it is one of the earliest and best-known examples of a non-distributive logic.
- 8.
Orthocomplemented modular lattices should not be confused with orthomodular lattices. The set of lattices in the latter category is a proper subset of those in the former.
- 9.
We cannot go into the details here, but we mention Anderson et al. (1992, Sect. 65) too.
- 10.
Another representation of modular lattices was obtained by Jónsson (1953). He proved that every lattice that can be represented with join being \(R_1;R_2;R_1\) (where \(R_1\) and \(R_2\) are two equivalence relations on a set) is modular.
- 11.
“Right” and “left” are, obviously, at hand, in particular, they are used in the theory of fields.
- 12.
Such functions, in an abstract setting, i.e., outside of Galois theory, have been studied by Everett (1944) and Ore (1944, 1962). Since the power set (or a set of special subsets of a set) has a natural ordering on it, namely, the subset relation, it is immediate that Galois connections on a collection of subsets induce a lattice (cf. Birkhoff 1967, V.8).
- 13.
Wille’s notions of (formal) context and (formal) concept, which he designed for computer science applications should not be confused with philosophical investigations of concepts following Wittgenstein or with the use of the term “concept” in cognitive science.
- 14.
See Birkhoff and Frink (1948, Sect. 11).
- 15.
- 16.
“Substructural logics” is often used as an honorific to include relevance logics such as \(\mathbf {T}\), \(\mathbf {E}\) and \(\mathbf {R}\) too, even though these logics have a distributive lattice reduct in their Lindenbaum algebra.
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Acknowledgements
We are grateful to the editors of the volume for inviting us to contribute a paper, and for providing a helpful report from the second reader. Our work on this paper was supported by the Insight Grant entitled “From the Routley–Meyer semantics to gaggle theory and beyond: The evolution and use of relational semantics for substructural and other intensional logics” (#435-2019-0331) awarded by the Social Sciences and Humanities Research Council of Canada.
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Bimbó, K., Dunn, J.M. (2022). St. Alasdair on Lattices Everywhere. In: Düntsch, I., Mares, E. (eds) Alasdair Urquhart on Nonclassical and Algebraic Logic and Complexity of Proofs. Outstanding Contributions to Logic, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-030-71430-7_12
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