Orders, Compatibility, Polarity
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Three roads to complete lattices: orders, compatibility, polarity Wesley H. Holliday University of California, Berkeley Preprint of December 2020. Forthcoming in Algebra Universalis. Abstract This note aims to clarify the relations between three ways of constructing complete lattices that appear in three different areas: (1) using ordered structures, as in set- theoretic forcing, or doubly ordered structures, as in a recent semantics for intuitionistic logic; (2) using compatibility relations, as in semantics for quantum logic based on ortholattices; (3) using Birkhoff’s polarities, as in formal concept analysis. Keywords: complete lattice, representation, closure operator, doubly ordered structure, compatibility, proximity, polarity, orthoframe, Boolean algebra, Heyting algebra, ortholattice, formal concept analysis MSC: 06B23, 06B15, 06C15, 06D20, 06D22, 03G05, 03G10, 03G12 1 Introduction Several approaches to the representation of complete lattices appear in applications of lattice theory to logic and computer science. These approaches include: (1) using ordered structures, as in set-theoretic forcing, or doubly ordered structures, as in a recent semantics for intuitionistic logic; (2) using compatibility relations, as in semantics for quantum logic based on ortholattices; (3) using Birkhoff’s polarities, as in formal concept analysis. The aim of this note is to clarify the relations between these three ways of constructing complete lattices. In each case, the relevant structure provides a closure operator c on the lattice of downsets of a preordered set or on the lattice of all subsets of a set. We are interested in the complete lattice of fixpoints of c, taking advantage of the first part of the following classic theorem (see, e.g., [7, Proposition 7.2(ii)], [6, Theorem 5.3]). Theorem 1.1. The fixpoints of a closure operator on a complete lattice form a complete lattice under the restricted lattice order. Conversely, any complete lattice is isomorphic to the lattice of fixpoints of a closure operator on a powerset lattice (resp. a downset lattice). 1 In Section 2, we review roads (1), (2), and (3) in turn. Then in Section 3, we connect the roads via direct transformations between each type of structure. We conclude in Section 4 with some questions about the tradeoffs of traveling down one road rather than another. We first fix some conventions. For an abstract lattice L, we denote its order by ≤. When we refer to a lattice of sets, the lattice order is always the inclusion order ⊆. Given a binary relation ≺ on a set X, we write y ≺ x for (y; x) 2 ≺, and given an expression ', we write 8y ≺ x ' for 8y(y ≺ x ) '), and 9y ≺ x ' for 9y(y ≺ x and '): 2 The three roads 2.1 Orders The first road begins with the following generalization of a preordered set. Definition 2.1. A doubly ordered structure is a triple (X; ≤1; ≤2) where X is a nonempty set and ≤1 and ≤2 are preorders on X. For Y ⊆ X, define: Inti(Y ) = fx 2 X j 8y ≤i x y 2 Y g; Cli(Y ) = fx 2 X j 9y ≤i x y 2 Y g; 0 00 0 00 c12(Y ) = Int1(Cl2(Y )) = fx 2 X j 8x ≤1 x 9x ≤2 x x 2 Y g: Y is a ≤i-downset if Y = Inti(Y ), and Down(X; ≤i) is the collection of all ≤i-downsets. Remark 2.2. Urquhart [18] uses the term ‘doubly ordered set’ for structures as in Defini- tion 2.1 in which for all x; y 2 X, if x ≤1 y and x ≤2 y, then x = y. We do not assume this condition, for the reason explained before Theorem 2.7 below. It is straightforward to check the following facts. Lemma 2.3. Every c12-fixpoint is a ≤1-downset. Proposition 2.4. For any doubly ordered structure (X; ≤1; ≤2), c12 is a closure operator on Down(X; ≤1). One can also observe that the functions Int1(X n ·): Down(X; ≤2) ! Down(X; ≤1) and Int2(X n ·): Down(X; ≤1) ! Down(X; ≤2) form an antitone Galois connection, so their composition Int1(X nInt2(X n·)) = Int1(Cl2(·)) = c12(·) is a closure operator on Down(X; ≤1). The following example is well known in the literature on forcing in set theory (see, e.g., [15]). Theorem 2.5. L is a complete Boolean algebra if and only if L is isomorphic to the lattice of c12-fixpoints of a preordered set, i.e., a doubly ordered structure in which ≤1 = ≤2. When ≤1 = ≤2, the c12-fixpoints are exactly the regular open sets in the topology on X whose open sets are all the ≤1-downsets. As observed by Tarski [16], the regular open sets of any space form a complete Boolean algebra (see, e.g., [11, Ch. 10]), which gives us the 2 right-to-left direction of Theorem 2.5. In the left-to-right direction, one may in fact take the preorder to be a partial order. The poset is constructed by deleting the bottom element of L and restricting the lattice order; the regular open downsets are then exactly the principle downsets plus ?, yielding an isomorphic copy of L. For Heyting algebras, we have the following analogue of Theorem 2.5 from [3] and [14], where it is used to give a semantics for intuitionistic logic based on doubly ordered structures. Theorem 2.6. L is a complete Heyting algebra if and only if L is isomorphic to the lattice of c12-fixpoints of a doubly ordered structure in which ≤2 ⊆ ≤1. Doubly ordered structures in which ≤2 ⊆ ≤1 were introduced by Fairtlough and Mendler [10], who observed that in this case c12 is a nucleus (inflationary, idempotent, and multiplicative operation) on the complete Heyting algebra of ≤1-downsets (in fact, Fairtlough and Mendler worked with upsets and therefore defined c12 in terms of ≥1 and ≥2 instead of ≤1 and ≤2 as in Definition 2.1). Since the fixpoints of a nucleus on a complete Heyting algebra again form a complete Heyting algebra (see, e.g., [13] or [9, p. 71]), this gives us the right-to-left direction of Theorem 2.6. Also note that in the left-to-right direction of Theorem 2.6, we may assume that ≤1 and ≤2 are partial orders (see [3, Proposition 4.5]). On the other hand, even the assumption that ≤1 and ≤2 are preorders is not necessary for Proposition 2.4, as discussed in Remark 2.12. Going beyond Boolean and Heyting algebras, Allwein and MacCaull [1] observed that by moving from Urquhart’s [18] notion of ‘doubly ordered set’ to the more general notion in Definition 2.1, one can represent arbitrary complete lattices. For comparison with later constructions, we include a proof of the following. Theorem 2.7. L is a complete lattice if and only if L is isomorphic to the lattice of c12- fixpoints of a doubly ordered structure. Proof. The right-to-left direction follows from Proposition 2.4 and Theorem 1.1. From left to right, define (X; ≤1; ≤2) as follows: 1. X = f(a; b) 2 L2 j a 6≤ bg; 2. (a; b) ≤1 (c; d) , a ≤ c; 3. (a; b) ≤2 (c; d) , b ≥ d. The elements of the form (a; 0) ordered by ≤1 form a lattice isomorphic to Lnf0g. Thus, the principal ≤1-downsets of elements of the form (a; 0), plus ?, ordered by ⊆, form a lattice isomorphic to L. Therefore, to prove the theorem it suffices to show that the c12-fixpoints are exactly the principal ≤1-downsets of elements of the form (a; 0), plus ?. First, we show that each principal ≤1-downset #1(a; 0) is a c12-fixpoint. Suppose that 0 0 (c; d) 62 #1(a; 0), so c 6≤ a. Then (c; a) 2 X and (c; a) ≤1 (c; d). Now consider any (c ; a ) ≤2 0 0 0 0 0 0 (c; a), so c 6≤ a and a ≥ a. Then c 6≤ a, so (c ; a ) 6≤1 (a; 0). Hence (c; a) 62 Cl2(#1(a; 0)), which with (c; a) ≤1 (c; d) implies (c; d) 62 c12#1(a; 0). W Suppose U = f(ai; bi) j i 2 Ig is a c12-fixpoint. Where e = fai j i 2 Ig, we claim that U = #1(e; 0). Clearly U ⊆ #1(e; 0). Since U is a ≤1-downset, to show U ⊇ #1(e; 0) it 3 suffices to show that (e; 0) 2 U. Since U is a c12-fixpoint, it suffices to show that for all (a; b) ≤1 (e; 0) there is a (c; d) ≤2 (a; b) such that (c; d) 2 U. Thus, suppose (a; b) ≤1 (e; 0), so a 6≤ b and a ≤ e. It follows that for some i 2 I, ai 6≤ b. For otherwise e ≤ b, which with a ≤ e implies a ≤ b, contradicting the fact that a 6≤ b. Hence (ai; b) 2 X and (ai; b) ≤2 (a; b). Finally, since U = f(ai; bi) j i 2 Ig is a ≤1-downset, we have (ai; b) 2 U, which completes the proof that (e; 0) 2 U. Theorems 2.5 and 2.6 can be viewed as showing that the doubly ordered structure used in the proof of Theorem 2.7 can be cut down in the Boolean and Heyting cases as follows: • if L is a Boolean algebra, then one may restrict X to the pairs (a; :a), where :a is the complement of a, in which case the restricted relations satisfy ≤1 = ≤2. • if L is a Heyting algebra, then one may define (a; b) ≤2 (c; d) if both a ≤ c and b ≥ d, in which case this modified relation satisfies ≤2 ⊆ ≤1.