Macneille Completion and Profinite Completion Can Coincide on Finitely Generated Modal Algebras

Following [3] we confirm a conjecture of Yde Venema by piecing together results from various authors. Specifically, we show that if A is a residually finite, finitely generated modal algebra such that HSP(A) has equationally definable principal congruences, then the profinite completion of A is the MacNeille completion of A, and ♦ is smooth. Specific examples of such modal algebras are the free K4-algebra and the free PDL-algebra. 1. Completions and topologies Let B = 〈B;∧,∨,¬, 0, 1〉 be a Boolean algebra. Given b ∈ B we write b↓= {a ∈ B | a ≤ b} (b↑ is defined dually). We say S ⊆ B is join-dense in B iff for every a ∈ B, a = ∨ (a↓∩S) (meet-density is defined dually). A completion of a lattice B is a pair (m,C), where m : B ↪→ C is a lattice embedding into a complete lattice C. Completions (m,C) and (k,D) of B are isomorphic if gm = k for some lattice isomorphism g : C → D. If (m,C) is a completion of B, let ρB be the topology on C generated by basis {[m(a),m(b)] | a, b ∈ B} (where [x, y] = {z ∈ C | x ≤ z ≤ y}). By γ B, γ ↑ B and γB we denote the Scott topology, the dual Scott topology, and the biScott topology on B respectively. Let AtB be the (possibly empty) set of atoms of B, and let Atω B be the set of all finite joins of atoms of B. Then ιB is the topology generated by the basis {[a,¬b] | a, b ∈ Atω B}. By [9, Section 2], ιB = γB if B is complete and atomic. The MacNeille completion [2] of a Boolean algebra B is defined up to isomorphism as a completion (m,C) such that m[B] is join-dense in C (by [5, Theorem V-27] C is then also a Boolean algebra). We denote the MacNeille completion of B by B̄. Alternatively [13, Theorem 4.5], B̄ can be characterized up to isomorphism as a completion (m,C) of B such that 〈C, ρB〉 is Hausdorff. If f : B → C is an order-preserving map between Boolean algebras, then f◦ : B̄ → C̄, defined by f◦ : x 7→ ∨ {f(a) | mB(a) ≤ x}, is the lower extension of f . The upper extension f• is defined dually. Alternatively [13, Section 5], f◦ is the (pointwise) largest (ρB, γ ↓ C̄)continuous extension of f , and f• is the smallest (ρB, γ ↑ C̄)-continuous extension of f . We say f is smooth if f◦ = f•. Given a modal algebra A = 〈A;♦〉, let ΦA := {θ ∈ ConA | A/θ is finite}. We say A is residually finite if for all a, b ∈ A with a 6= b, there exists θ ∈ ΦA such that a/θ 6= b/θ. The inverse system 〈{A/θ}θ∈ΦA , fθψ〉, where fθψ : A/θ A/ψ (for all θ, ψ ∈ ΦA such that θ ⊆ ψ) is defined by fθψ : a/θ 7→ a/ψ, has a projective limit  = { α ∈ ∏ ΦA A/θ | ∀θ, ψ ∈ ΦA with θ ⊆ ψ, if α(θ) = a/θ then α(ψ) = a/ψ } . Date: November 7, 2007. This research was supported by VICI grant 639.073.501 of the Netherlands Organization for Scientific Research (NWO).