A locally finite characterization of AE(0) and related classes of compacta

A locally finite characterization of AE(0) and related classes of compacta David Milovich Texas A&M International University david.milovich@tamiu.edu http://www.tamiu.edu/ dmilovich/ March 13, 2014 Spring Topology and Dynamics Conference University of Richmond 1 / 9

Stone duality notation A compactum is a compact Hausdorff space. A boolean space is a compactum with a clopen base. Clop is the contravarient functor from boolean spaces and continuous maps to boolean algebras and homomorphisms. Clop(X ) is ({K X : K clopen},,, K X \ K}). Clop(f )(K) = f 1 [K]. Modulo isomorphism, the inverse of Clop is the functor Ult: Ult(A) is {U A : U ultrafilter} with clopen base {{U Ult(A) : a U} : a A}; Ult(φ)(U) = φ 1 [U]. 2 / 9

Open is dual to relatively complete. A boolean subalgebra A of B is called relatively complete if every b B has a least upper bound in A. Let A rc B abbreviate A is relatively complete in B. A boolean homomorphism φ: A B is called relatively complete if φ[a] rc B. A boolean homomorphism φ is relatively complete iff Ult(φ) is open. 3 / 9

AE(0) spaces Definition A boolean space X is an absolute extensor of dimension zero, or AE(0) for short, if, for every continuous f : Y X with Y Z boolean, f extends to a continuous g : Z X. 4 / 9

AE(0) spaces Definition A boolean space X is an absolute extensor of dimension zero, or AE(0) for short, if, for every continuous f : Y X with Y Z boolean, f extends to a continuous g : Z X. Given a boolean space X of weight κ, the following are known to be equivalent: X is AE(0). X is Dugundji, i.e., a retract of 2 κ. X 2 κ = 2 κ. There exists Y such that X = Y 2 κ and, for all α < β < κ, the projection π α,β : Y β Y α is open. Clop(X ) has an additive rc-skeleton, i.e., if n < ω, θ is a regular cardinal, and Clop(X ) N i H(θ) for all i < n, then Clop(X ) i<n N i rc Clop(X ). 4 / 9

Multicommutativity A poset diagram of boolean spaces is pair of sequences ( X, f ) with dom( X ) a poset, Xi a boolean space for all i dom( X ), fj,i : X i X j continuous for all j < i, and f k,i = f k,j f j,i for all k < j < i. Given a poset diagram ( X, f ) and I dom( X ), let { lim(x i : i I ) = p } X i : {j < i} I p(j) = f j,i (p(i)). i I Call a poset diagram ( X, f ) multicommutative if, for all i dom( X ), j<i f j,i maps X i onto lim(x j : j < i). 5 / 9

A new characterization of AE(0) A poset P is called locally finite if every lower cone is finite. A poset diagram ( X, f ) is locally finite if dom( X ) is locally finite and every X i is finite. A locally finite poset is a lattice iff every nonempty finite subset has a least upper bound. A poset diagram ( X, f ) is called a lattice diagram if dom( X ) is a lattice. Theorem Given a boolean space X, the following are equivalent. X is AE(0). X is homeomorphic to the limit of a multicommutative locally finite poset diagram. X is homeomorphic to the limit of a multicommutative locally finite lattice diagram. 6 / 9

Long ω 1 -approximation sequences For every ordinal α, let α = max{β α : β < ω 1 or γ α γ = β}; α = α + [α]; 7 / 9

Long ω 1 -approximation sequences For every ordinal α, let α = max{β α : β < ω 1 or γ α γ = β}; α = α + [α]; [α]0 = α; [α]n+1 = [[α] n ]; α n = i<n [α] i ; 7 / 9

Long ω 1 -approximation sequences For every ordinal α, let α = max{β α : β < ω 1 or γ α γ = β}; α = α + [α]; [α]0 = α; [α]n+1 = [[α] n ]; α n = i<n [α] i ; ℸ(α) = min{n < ω : [α] n = 0}. If 1 k < ω and α ω k, then ℸ(α) k. Given θ regular and uncountable, a long ω 1 -approximation sequence is a transfinite sequence (M α ) α<η of countable elementary substructures of H(θ) such that (M β ) β<α M α for all α < η. (Milovich, 2008) If M is a long ω 1 -approximation sequence and α, β dom( M), then Mβ M α β α M α M β M α ; for all i < ℸ(α), M i α = {M γ : α i γ < α i+1 } is a directed union; hence, Mα i H(θ). 7 / 9

n-commutativity Call a lattice diagram ( X, f ) n-commutative if, for all i dom( X ) and all j 0,..., j n 1 < i, k K f k,i maps X i onto lim(x k : k K) where K = m<n {k : k j m}. Call a boolean space n-commutative if it is homeomorphic to the limit of an n-commutative locally finite lattice diagram. The Stone dual of 2-commutative boolean space has been studied under the name of strong Freese-Nation property. There are 2-commutative boolean spaces of weight ℵ 2 that are known to not be AE(0), e.g., the symmetric square of 2 ω 2. (Milovich) Every locally finite lattice of size ℵ n 1 contains a cofinal suborder that is an n-ladder, i.e., a locally finite lattice in which every element has at most n maximal strict lower bounds. 8 / 9

The strong Freese-Nation property is strictly stronger A boolean algebra A is said to have the Freese-Nation property, or FN, if A M rc A whenever A M H(θ). 9 / 9

The strong Freese-Nation property is strictly stronger A boolean algebra A is said to have the Freese-Nation property, or FN, if A M rc A whenever A M H(θ). Heindorf and Shapiro introduced the strong Freese-Nation property, or SFN, and showed that it implied the FN, and asked if the implication was strict. If A is a boolean algebra, (M α ) α< A is a long ω 1 -approximation sequence, and A M 0, then, for all α < A, i < ℸ(α), and a A M α \ β<α M β, set σ i (a) = min{b A M i α : b a} if it exists. (Milovich) A has the FN iff, for all M, α, i, a as above, σi (a) exists. (Milovich) A has the SFN iff, for all M as above, A = (A,,,, σ 0, σ 1, σ 2,...) is a locally finite partial algebra, i.e., every finite subset of A is contained in a finite subalgebra of A. (Milovich) There is a boolean algebra of size ℵ 2 that has the FN but not the SFN. 9 / 9