Median algebra with a retraction: an example of variety closed under natural extension

Median algebra with a retraction: an example of variety closed under natural extension

Bruno Teheux

(joint work with Georges Hansoul)

The Big Picture

Claims about natural extensions

Natural extension provides

‘canonical extension’ for non lattice-based algebras ; insight about the construction of canonical extension.

Back to the roots : canonical extension

L=hL,∨,∧,0,1iis a DL

Canonical extensionL^δofLcomes with topologiesιandδ:

I L^δ is doubly algebraic.

I L,→L^δ.

I Lis dense inL^δ_ι.

I Lis dense and discrete inL^δ_δ.

Canonical extension comes with a tool to extend maps

Problem.Givenf:L→E, definef^δ:L^δ→E^δ.

Solution.

I Lis made of the isolated points ofL^δ,

I Lis dense inL^δ_δ,

I f^δ:=lim inf_δf and f^π :=limsup_δf.

Leads to canonical extension of ordered algebras :

Jónsson-Tarski (1951), Gehrke and Jónsson (1994), Dunn, Gehrke and Palmigiano (2005), Gehrke and Harding (2011), Gehrke and Vosmaer (2011), Davey and Priestley (2011). . .

Why canonical extension ?

It provides completeness results formodal logicswith respects to classes of KRIPKEframes :

Jónsson, B. (1994). On the canonicity of Sahlqvist identities.Studia Logica, 53(4), 473–491.

Gehrke, M., Nagahashi, H., and Venema, Y. (2005). A Sahlqvist theorem for distributive modal logic.Ann. Pure Appl. Logic, 131(1-3), 65–102.

Hansoul, G., and Teheux, B. (2013). Extending Łukasiewicz logics with a modality : algebraic approach to relational semantics.Studia Logica, 101(3), 505–545.

Is it possible to generalize canonical extension to non lattice-based algebras ?

Problem 1.Define the natural extensionA^δofA:

Davey, B. A., Gouveia, M., Haviar, M., and Priestley, H. (2011).

Natural extensions and profinite completions of algebras.Algebra Universalis, 66, 205–241.

Problem 2.Extendf:A→B to f^δ:A^δ→B^δ. We give apartial solution.

Natural extension of algebras

The framework of natural extension

A^δ can be defined ifAbelongs to some ISP(M)

whereMis class of finite algebras of the same type.

A^δ can more easily be computed ifISP(M)is dualisable (in the sense ofnatural dualities).

We adopt the setting of natural dualities

M≡a finite algebra

A discrete alter-ego topological structureM A∈ISP(M) e

Algebra Topology

M M

A=ISP(M) X =ISecP⁺(M e ) A A^∗=A(A,M)≤_c M

e

A

X e

∗=X(X e

,M e

)≤M^X X

e

Definition.M e

yields a natural duality forISP(M)if (A^∗)∗ ' A, A∈ISP(M).

Natural extension of an algebra can be constructed from its dual

Priestley duality is a natural duality :L'(L^∗)∗

Proposition(Gehrke and Jónsson)

IfL∈DLthenL^δis the algebra oforder-preserving mapsfrom L^∗to 2

e .

Assume thatM e

yields a dualityforISP(M).

Proposition(Davey and al.)

IfA∈ISP(M), thenA^δis the algebra ofstructure-preserving mapsfromA^∗ toM

e .

Natural extension of median algebras

The variety of median algebras is an old friend. . .

The expression

(x∧y)∨(x∧z)∨(y∧z)

defines an operationm≤(x,y,z)on a distributive lattice(L,≤).

Definition.(Avann, 1948)

median algebra A= (A,m) ⇐⇒ subalgebra of some(L,m≤)

Example.Set2:=h{0,1},miwheremis the majority function.

Theorem.The varietyA_m of median algebras isISP(2).

(x∧y)∨(x∧z)∨(y∧z) Examples.

m(•,•,•) =•

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Median graphs Some metric spaces

. . . in which semilattice orders can be defined

For everya∈A, the relation ≤_a defined onAby b≤_ac if m(a,b,c) =b.

is a∧-semilattice order onAwithb∧_ac =m(a,b,c).

Semillatices obtained in this way are themedian semilattices.

Proposition.In a median semilattice, principal ideals are distributive lattices.

Grau (1947), Birkhoff and Kiss (1947), Sholander (1952, 1954), Isbell (1980), Bandelt and Hedlíková (1983). . .

There is a natural duality for median algebras

A_m=ISP(2) 2

e

:=h{0,1},≤,·^•,0,1, ιi.

Theorem(Isbell (1980), Werner (1981)). The structure 2 e

yields a logarithmicduality forA_m.

A^δ is the algebra of structure-preserving maps x: A^∗ →2 e .

Natural extension completes everything it can complete

Theorem.Leta∈A.

I hA^δ,≤_aia bounded-complete extension ofhA,≤_ai.

I IfIis a distributive lattice inAthencl_A^δ

ι(I) =I^δ

Natural extension of maps

A can be defined topologically in A

X_p(A^∗,M e

)≡set of morphisms defined on a closed substructure ofA^∗.

Definition.

O_f :={x ∈ X(A^∗,M e

)|x ⊇f}, f ∈ X_p(A^∗,M e )

∆ :={O_f |f ∈ X_p(A^∗,M e

)}

Working assumption. M e

yields a full logarithmic duality for ISP(M)andM

e

is injective inIScP⁺(M e ).

Proposition.

I ∆is a basis of topologyδ

I Ais dense and discrete inA^δ.

I In the settings of DL, we get the known topology.

We canonically extends maps to multi-maps

Input :

f :A→B

Output :

f⁺:A^δ →Γ(B^δ_ι)

We canonically extends maps to multi-maps

Input :

f :A→B

Output :

f⁺:A^δ →Γ(B^δ_ι)

The multi-extension of f : A → B

Intermediate step :Consider

¯f :A→Γ(B^δ_ι) :a7→ {f(a)}.

Recall thatAis dense inA^δ_δ andΓ(B^δ_ι)is a complete lattice.

Definition.Themulti-extension f⁺of f is defined by f⁺:A^δ_δ →Γ(B_ι^δ) :x 7→limsup_δ¯f(x), In other words,

f⁺(x) =\ {cl_Bδ

ι(f(A∩V))|V ∈δx}, f⁺(x)

F =\

{f(A∩V)

F |V ∈δx}, F bB^δ

The multi-extension is a continuous map

Definition.

We say thatf is smooth if#f⁺(x) =1 for allx ∈A^δ.

Letσ↓be theco-Scotttopology onΓ(B^δ_ι).

Proposition.

I f⁺is the smallest(δ, σ↓)-continuous extension fromA^δ_δto Γ(B^δ_ι).

I f is smooth if and only if it admits an(δ, ι)-continuous extensionf^δ :A^δ →B^δ satisfyingf^δ(x)∈f⁺(x).

This construction sheds light on canonical extension

Proposition.Iff:A→Bis a map between DLs with lower extensionf^δ and upper extensionf^π, then for anyx ∈L^δ

f^δ(x) =^ f⁺(x), f^π(x) =_

f⁺(x).

Natural extension of median algebras with a

retraction.

Natural extension of expansions of median algebras

General framework.

Let

A=hA,m,r,ai wherehA,mi ∈ A_m,a∈Aandr:A→A

Set

r^δ(x) = ∧_ar⁺(x), x ∈ hA,mi^δ. A^δ :=hhA,mi^δ,r^δ,ai

Natural properties

Definition.A propertyPof algebras inAis natural if A|=P =⇒ A^δ |=P, A∈ A

Example.The property ‘being a median algebra of a Boolean algebra’ is natural.

Median algebras with a retraction

Definition.An idempotent homomorphismr:A→Asuch that u(A)is convex is called aretraction.

Proposition.A mapr:A→Ais a retraction if and only if r(m(x,y,z)) =m(x,r(y),r(z)), x,y,z ∈A.

Definition.An algebrahA,m,r,aiis a pointed retract algebraif r is a retraction of the median algebrahA,mianda∈A.

The variety of pointed retract algebras is natural

Theorem.IfAis a pointed retract algebra thenA^δis a pointed retract algebra.

Sketch of the proof.

Proves equalities of the type

(r ◦m)^δ=r^δ◦m using continuity properties of the extensions.

The variety of pointed algebras with operator is natural.

Definition.An algebraA=hA,m,f,aiis a pointed median algebra with operator ifhA,miis a median algebra,a∈Aand

f(m(a,x,y)) =m(a,f(x),f(y)), x,y ∈A.

TheoremLethA,m,f,aibe a pointed median algebra with operator.

I f is smooth.

I A^δis a pointed median algebra with operator.

Sketch of the proof.

f can be dualized as a relationR onA^∗ andf^δcan be explicitly computed withR.

Questions/Problems.

I Interesting instances of natural extensions of maps (in non-ordered based algebras).

I Successful applications of the whole theory.

I Find canonical (continuous) way to pick-up some element inf⁺(x).

I Intrinsic definition ofδin the non-dualizable setting.