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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