sheafrepresentationsofmv-algebras...
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Sheaf representations of MV-algebras via Stone duality
Sheaf representations of MV-algebras
via Stone duality
Mai Gehrke(joint work with Sam van Gool and Vincenzo Marra)
CNRS and Universite Paris Diderot (Paris VII)
5 August 2013
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Sheaf representations of MV-algebras via Stone duality
MV-algebras
History of MV-algebras
◮ MV-algebras were introduced by C.C. Chang as algebraicsemantics of Lukasiewicz infinite-valued logic (a propositionalcalculus with truth values in [0, 1])
◮ MV-algebras have been studied extensively, mainly by Mundiciand co-workers. They have developed the structure theory aswell as links with other areas and applications
◮ The category of MV-algebras is equivalent to the category ofunital lattice-ordered abelian groups
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Sheaf representations of MV-algebras via Stone duality
MV-algebras
MV-algebrasAn MV-algebra is an algebra (A,⊕,¬, 0) such that
◮ (A,⊕, 0) is a commutative monoid,
◮ ¬¬x = x, that is, ¬ is an involution,
◮ x⊕ 1 = 1 where 1 := ¬0,
◮ ¬(¬x⊕ y)⊕ y = ¬(¬y ⊕ x)⊕ x.
◮ MV-algebras are bounded distributive lattices in theterm-definable operations:
a ∨ b := ¬(¬a⊕ b)⊕ b,
a ∧ b := ¬(¬a ∨ ¬b).
A variety of residuated lattices with the property that x⊕ x = ximplies A is a BA
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Sheaf representations of MV-algebras via Stone duality
MV-algebras
Examples of MV-algebras
◮ Boolean algebras
◮ The unit interval [0, 1] in its natural order is an MV-algebrawith
a⊕ b = min{a+ b, 1}, ¬a = min{1− a}
◮ The MV-algebra C(X, [0, 1]) of continuous functions from aspace X to [0, 1] with point-wise operations
◮ Free MV-algebras, consisting of the McNaughton functions onthe unit cube
◮ Nonstandard extensions of [0, 1] (ultrapowers) are examples ofMV-algebras with infinitesimal elements
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Sheaf representations of MV-algebras via Stone duality
MV-algebras
Representation theory for MV-algebras
◮ Duality between finitely presented MV-algebras and rationalpolyhedra [Marra and Spada 2012]
◮ Priestley duality for MV-algebras [N. G. Martinez 1994;Martinez- H. Priestley 1998; G-Priestley 2007-08](Is it useful for MV-algebraists?)
◮ Sheaf representations over the maximum MV-spectrum[Keimel 1968; Filipoiu-Georgescu 1995] and over theMV-spectrum [Kennison 1976; Cornish 1980; Yang 2006;Dubuc-Poveda 2010]
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Sheaf representations of MV-algebras via Stone duality
The different dual spaces of an MV-algebra
The lattice spectrumLet D be a distributive lattice, the points, x ∈ XD, of the latticespectrum of D are in one-to-one correspondence with each of thefollowing:
◮ hx : D → 2 a lattice homomorphism
◮ Ix a prime ideal of D (= h−1(0))
◮ Fx a prime filter of D (= h−1(1))
If, in addition, D is finite, then the above are equivalent to
◮ Fx = ↑p where p ∈ J(D) and Ix = ↓m where m ∈ M(D)
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Sheaf representations of MV-algebras via Stone duality
The different dual spaces of an MV-algebra
Stone’s representation theorem and dualityLet D be a distributive lattice, XD the spectrum of D, then
η : D → P(XD)
a 7→ a = {x ∈ XD | hx(a) = 1}
is an injective lattice homomorphism
Topologies on XD:
◮ τD = <a | a ∈ D> (spectral or Stone topology)
◮ τ∂D = <(a)c | a ∈ D> (dual spectral topology)
◮ πD = <a, (a)c | a ∈ D> (Priestley topology)
In all three Im( ) can be characterized (order-)topologically
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Sheaf representations of MV-algebras via Stone duality
The different dual spaces of an MV-algebra
The MV-spectrum
A simple but important fact in the theory of MV-algebras is that
θ : A −→ Con(A)
a 7−→ θ(a) = <(0, a)>Con(A)
is a bounded lattice homomorphism.
The image of this map is the lattice Confin(A) of finitelygenerated MV-algebra congruences of A (and thus thesecongruences are pairwise permuting).
The MV-spectrum of A, is the dual space, Y , of Confin(A)
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Sheaf representations of MV-algebras via Stone duality
The different dual spaces of an MV-algebra
The MV-spectrum
Since A −։ Confin(A) is a bounded distributive lattice quotient,by duality, Y → X may be seen as a closed subspace of X
The MV-spectrum may also be seen as the set of those MV-ideals(non-empty downsets closed under ⊕) that are prime in the sensethat one of a⊖ b(:= ¬(¬a⊕ b)) and b⊖ a is a member for alla, b ∈ A. This is the same set Y ⊆ X.
The spectral topology on Y is also the hull-kernel or spectraltopology corresponding to the MV-ideals of A.
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Sheaf representations of MV-algebras via Stone duality
The different dual spaces of an MV-algebra
Relative normality of the dual of the MV-spectrum
◮ The following are equivalent:
◮ A bounded distributive lattice D is normal: For all a, b ∈ D, ifa ∨ b = 1 then there are c, d ∈ A with c ∧ d = 0 and a ∨ d = 1and c ∨ b = 1
◮ Each point in the dual space of D is below a unique maximalpoint
◮ The inclusion of the maximal points of the dual space of Dadmits a continuous retraction
◮ For any MV-algebra A, the lattice Confin(A) is relativelynormal (that is, each interval [a, b] is a normal lattice)
As a consequence Y is always a root-system, that is, ↑y is a chainfor each y ∈ Y
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Sheaf representations of MV-algebras via Stone duality
The different dual spaces of an MV-algebra
The maximal MV-spectrum
Given an MV-algebra, A, the subspace Z of Y of maximalMV-ideals of A is called the maximal MV-spectrum
Since Confin(A) is relatively normal for any MV-algebra Y is aroot-system and the map
m : Y −→ Z
y 7→ unique maximal point above y
is a continuous retraction. The maximal MV-spectrum is compactHausdorff, but not in general spectral
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Sheaf representations of MV-algebras via Stone duality
Duality for ⊕ and ¬
Extended Priestley duality for MV -algebras(from BLAST’08 on [G-Priestley’07-’08])
The dual spaces of MV -algebras are the spaces (X,6, τ, ·)satisfying:
◮ (X,6, τ) is a Priestley space with bounds
◮ (X, τ,6, ·, 1) is an ordered topological monoid
◮ The · is open, commutative, and has a lower adjoint
◮ The element 0 is absorbant
◮ ∀x, y [ x 6= y ⋆ x ⇒ ∀z (x 6 z or y · z 6 y ⋆ x )] wherey ⋆ x is the least z ∈ X such that
∀x′ [y � x′ ⇒ x′ · x 6 z]
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Sheaf representations of MV-algebras via Stone duality
Duality for ⊕ and ¬
A bit of explanation
To best make sense of this we need canonical extension, but thereis not enough time
◮ On an MV-algebra we have ⊕ and ¬, but also ⊖, and ⊙ and→, which are de Morgan duals of ⊕ and ⊖
◮ ⊖ is lower adjoint to ⊕◮ ⊙ is lower adjoint to →
◮ In [G-Priestley] we took → as basic. It is witnessed dually by⊙, which by DQA restricts to X ∪ {0}
◮ In order to get a First-Order characterization, we also needed⋆ (which plays no role in this work)
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Sheaf representations of MV-algebras via Stone duality
Duality for ⊕ and ¬
What we need here
◮ For lattice ideals I and J
I⊕J = ↓{a⊕ b | a ∈ I, b ∈ J}
◮ For x, x′ ∈ X we have Ix⊕Ix′ is either prime or all of A
◮ Define a partial operation on X by x+ x′ = z iff Ix⊕Ix′ = Iz
◮ The partial operation + has domain {(x, y) | i(x) > y}, wherei is the involution dual to ¬
This partial operation + witnesses fully the MV-algebra structureof A relative to its lattice reduct
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Sheaf representations of MV-algebras via Stone duality
The map k [G-van Gool-Marra]
Relation to the MV-spectrum
For each x ∈ X we have that x+ x is defined and
x ∈ Y ⇐⇒ x+ x = x
For each x ∈ X, there is a largest x′ ∈ X with
x+ x′ 6 x
and this x′ satisfies x′ + x′ = x′
This defines a retraction k : X → Y , which is continuous withrespect to the Priestley topology on X and the spectral topologyon Y . A function named k (defined differently, but with the sameaction) was present in Martinez’ work
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Sheaf representations of MV-algebras via Stone duality
The map k [G-van Gool-Marra]
Interpolation Lemma
NB! The map k is neither order preserving nor reversing
However it satisfies the following Interpolation Lemma:
If x 6 x′ then there is x′′ with
x 6 x′′ 6 x′ and k(x′′) > k(x) and k(x′′) > k(x′)
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Sheaf representations of MV-algebras via Stone duality
The map k [G-van Gool-Marra]
From X to Z with m ◦ kCombining the two earlier retractions we get
m ◦ k : (X,π) −→ (Z, τ)
The kernel of this map is given by the relation x1Wx2 iff there arex′1, x
′
2, x0 ∈ X with
x1
x′1
x0
x′2
x2
Proof: If mk(x1) = mk(x2), then take x′i = k(xi) andx0 = mk(xi).
For the converse note that if x 6 x′, then by (Int) there is x′′
between with greater k-image than both, but thenmk(x) = mk(x′′) = mk(x′). So all the elements of X in one ordercomponent have the same mk-image
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Sheaf representations of MV-algebras via Stone duality
A Kaplansky theorem for MV [G-van Gool-Marra]
Kaplansky’s theorem
[Kaplansky 1947]Let Z1, Z2 be compact Hausdorff spaces such that the latticesC(Z1, [0, 1]) and C(Z2, [0, 1]) are isomorphic. Then Z1 and Z2 arehomeomorphic spaces.
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Sheaf representations of MV-algebras via Stone duality
A Kaplansky theorem for MV [G-van Gool-Marra]
Kaplansky theorem for arbitrary MV-algebras
TheoremIf A1 and A2 are MV-algebras having isomorphic lattice reducts,then the max MV-spectra of A1 and A2 are homeomorphic.
◮ Note that the max MV-spectrum of an MV-algebra of theform C(Z, [0, 1]) is Z so that our result generalizesKaplansky’s result.
Proof (sketch).
The maximal MV-spectrum can be constructed from the latticespectrum by taking the topological quotient w.r.t. the relation Wand W is definable from just the order structure of X
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Sheaf representations of MV-algebras via Stone duality
Sheaf representations [G-van Gool-Marra]
Spectral sumWe have seen that there is a continuous retraction
k : (X,π) −→ (Y, τ)
It follows that k−1(↑y), call it Xy, is a closed subspace of X in thePriestley topology. In fact, Xy is a chain and
Xy = k−1(↑y) is the dual of the MV quotient Ay = A/Iy
Moreover, one can show (Patch):
For any finite cover (Ui)ni=1 of Y by τ∂-open sets, and any
collection (ai)ni=1 of clopen downsets of X such that
ai ∩ k−1(Ui ∩ Uj) = aj ∩ k−1(Ui ∩ Uj) (*)
holds for any i, j ∈ {1, . . . , n}. Then the set⋃n
i=1(ai ∩ q−1(Ui)) isa clopen downset in X
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Sheaf representations of MV-algebras via Stone duality
Sheaf representations [G-van Gool-Marra]
Patching property
X
k
Y
a1
a2
U1 U2
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Sheaf representations of MV-algebras via Stone duality
Sheaf representations [G-van Gool-Marra]
Patching property
X
k
Y
a1
a2
∃!b
U1 U2y
Xy
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Sheaf representations of MV-algebras via Stone duality
Sheaf representations [G-van Gool-Marra]
Transparent solution via duality
◮ There is at most one b ∈ A,
◮ It must satisfy b =⋃n
i=1(ai ∩ k−1(Ui)) =: K, and K is closed
◮ Prove that K is a open: Kc =⋃n
i=1(aic ∩ k−1(Ui))
◮ Prove that K is a downset: Interpolation Lemma!!!
◮ This also yields a formula for b by compact approximation:
b =
n∨
i=1
(ai ⊙ ¬mui),
where m,n ∈ N and ui ∈ A such that uic = Ui.
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Sheaf representations of MV-algebras via Stone duality
Sheaf representations [G-van Gool-Marra]
From spectral sums to sheaves
TheoremLet D be a distributive lattice with dual space X. Suppose thatq : (X,π) → (I, ρ) is a continuous surjection onto a stablycompact space1 which satisfies the property (Patch) as givenearlier. Then the associated etale space p : (E, σ) → (I, ρ∂) 1 is asheaf representation of D over Y
◮ We saw above that k : (X,π) → (Y, τ) satisfies the hypothesisof the above theorem
◮ One can also show that m ◦ k : (X,π) → (Z, τ) satisfies thehypothesis of the above theorem
1See Sam van Gool’s talk tomorrow morning23 / 24
Sheaf representations of MV-algebras via Stone duality
Sheaf representations [G-van Gool-Marra]
Sheaf decompositions of MV-algebras
As a consequence of the sum decompositions we obtain:
TheoremAny MV-algebra is representable as the global sections of a sheaf
1. over the maximal MV-spectrum, which is a compactHausdorff space, with stalks that are local MV-algebras (i.e.having a unique maximal ideal)
[Keimel 1968; Filipoiu-Georgescu 1995]
2. over the MV-spectrum with the dual spectral topology withstalks that are MV-chains
[Kennison 1976; Cornish 1980; Yang 2006; Dubuc-Poveda 2010]
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Sheaf representations of MV-algebras via Stone duality
Sheaf representations [G-van Gool-Marra]
Conclusion
We have obtained the sheaf representations of MV-algebras overtheir prime and maximal MV-spectra from a correspondingdecomposition of their Priestley dual spaces. This allows a directtreatment of all MV-algebras using only simple facts from thetheory of MV-algebras
If sheaf representations are interesting to MV-algebraists, then sois Priestley duality
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