A ring-theoretic characterisation of Oz spaces

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A ring-theoretic characterisation of Oz spaces

Oghenetega Ighedo

Department of Mathematical Sciences

University of South Africa (UNISA) Talk given at

STW-2013 ( São Sebãstiao, Brazil )

12th August 2013 ( joint work with Themba Dube )

Oghenetega Ighedo (UNISA) Oz-spaces 1 / 16

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Ozspaces

A subsetS of a topological spaceX isz-embeddedinX in case each zero-set ofSis the restriction toSof a zero-set ofX.

In the article

Spaces in which special sets arez-embedded,Can. J. Math.,28(1976), 673–690,

Blair calls a Tychonoff spaceX anOz spaceif every open set ofX is z-embedded.

A useful characterisation is thatX is anOzspaceif and only if every regular-closed subset ofX is a zero-set.

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Ozspaces

In the article

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Ozspaces

In the article

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Frames

Aframeis a complete latticeLin which the distributive law a∧_

S=_

{a∧x |x ∈S}

holds for alla∈LandS⊆L.

Frame homomorphismsare maps that preserve the frame structure.

An example of a frame is the lattice of open subsetsOX of a topological spaceX.

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Frames

S=_

{a∧x |x ∈S}

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Frames

S=_

{a∧x |x ∈S}

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Frames

An elementaofLisrather belowan elementb, writtena≺b, in case there is an elements, called aseparatingelement, such that a∧s =0ands∨b=1.

The frameLisregularifa=W

{x ∈L|x ≺a}for eacha∈L.

An elementaiscompletely belowb, writtena≺≺b, if there are elements(x_r)indexed by rational numbersQ∩[0,1]such that a=x₀,x₁=bandx_r ≺x_s forr <s.

The frameLiscompletely regularifa=W

{x ∈L|x ≺≺a}for eacha∈L.

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Frames

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Frames

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Frames

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Frames

CozLis the cozero part ofL, and is theregular sub-σ-frame consisting of all the cozero elements ofL.

βLis theStone- ˇCech compactificationofLand it is the frame of regular ideals ofCozL. We denote byj_L:βL→Lthe join map J 7→W

J, and the right adjoint ofj_Lis here denoted byr_L. By apointof a frame we mean a prime element, that is, an elementp<esuch that for anyaandb in the frame,a∧b≤p impliesa≤p orb≤p. We denote byPt(L)the set of all points of L.

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Frames

βLis theStone- ˇCech compactificationofLand it is the frame of regular ideals ofCozL.We denote byj_L:βL→Lthe join map J 7→W

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Frames

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Frames

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Frames

An elementaof a frameLis calledLindelöfwhenevera=W S impliesa=W

T for some countableT ⊆S, andLis Lindelöf whenevere∈Lis Lindelöf.

For any sublatticeAof a frameL, an idealJ ⊆Ais called

1 σ-properifW

S6=efor any countableS⊆J, and

2 completely properifW J 6=e,

the joins understood inL.

A frameLisrealcompactif anyσ-propermaximal ideal inCozLis completely proper.

A topological spaceX isrealcompactiff it can be embedded as a closed subset of a product of copies of the real lineR.

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Frames

1 σ-properifW

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Frames

1 σ-properifW

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Frames

1 σ-properifW

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Frames

1 σ-properifW

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Frames

1 σ-properifW

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Ozframes

Oz framesare the natural pointfree counterpart ofOz spaces.

B. Banaschewski, T. Dube, C. Gilmour, J. Walters-Wayland,Oz in pointfree topology

Quaestiones Mathematicae32(2009), 215–227.

They define a frameLto be anOz frameif everya∈Lis coz-embedded.

Example

Every Boolean frameLisOz.

IfL=OX, thenLis anOz frame iffX is anOz space.

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Ozframes

Example

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Ozframes

Example

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Ozframes

Example

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Ozframes

Example

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Ozframes

Example

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Pointfree ringRL

Throughout, by “ring" we mean a commutative ring with identity. For a ringAanda∈A, we let letMax(A)denote the set of maximal ideals of A. We set

M(a) ={M∈Max(A)|a∈M} and M(a) =\ M(a).

In the article

G. Mason,z-Ideals and Prime Ideals,J. Alg.26(1973), 280-297,

Mason calls an idealIofAaz-idealif for anyaandb inA, a∈I and M(a) =M(b) ⇒ b ∈I.

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In the article

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In the article

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In the article

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LetLbe a completely regular frame, andRL be the ring of real-valued continuous functions onLwithR^∗Las the subring of its bounded elements.

ForI∈βL,

M^I ={α∈ RL|r_L(cozα)⊆I} and O^I ={α∈ RL|r_L(cozα)≺I}.

For anya∈Lwe abbreviateM^r^L^(a)asM_a, and remark that

Ma={α∈ RL|cozα≤a}.

The maximal ideals ofRLare precisely the idealsM^I,forI∈Pt(βL).

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ForI∈βL,

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ForI∈βL,

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ForI∈βL,

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ForI∈βL,

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It is shown in

T. Dube,Contracting the socle in rings of continuous functions, Rend. Sem. Mat. Univ. Padova,123(2010), 37–53.

that annihilator ideals inRLare precisely the idealsMa^∗, fora∈L. In particular, for anyα∈ RL, Ann(α) =M_(cozα)^∗.

For any Tychonoff spaceX, the ringsC(X)andR(OX)are isomorphic.

B. Banaschewski,The real numbers in pointfree topology,

Textos de Matemática Série B, Departamento de Matemática da Universidade de Coimbra, 1997.

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It is shown in

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It is shown in

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It is shown in

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It is shown in

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Characterisation

An ideal of a ringAis aprincipalz-idealif it is of the formM(a)for somea∈A.

Corollary

An ideal ofRLis an intersection of maximal ideals iff it is of the form M^I, for someI∈βL.

In the ringRL, principalz-idealshave the following description.

Lemma

Principal z-ideals ofRL are precisely the idealsM_c for c ∈CozL.

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Corollary

Lemma

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Corollary

Lemma

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Corollary

Lemma

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Proof.

We recall thatM(α) =T

{M ∈Max(RL)|α∈M}.

PutP ={I∈Pt(βL)|α∈M^I}, then

P ={I∈Pt(βL)|r_L(cozα)≤I}.

SinceβLis spatial, VP =V

{I∈Pt(βL)|r_L(cozα)≤I}=r_L(cozα). So,

M(α) = \

{M∈Max(RL)|α∈M}

= \

{M^I |I∈ P}

= M^∧Pand by the Corollary,

= M^r^L^(coz^α)

= M_(cozα).

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Proof.

M(α) = \

= \

{M^I |I∈ P}

= M^r^L^(coz^α)

= M_(cozα).

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Proof.

M(α) = \

= \

{M^I |I∈ P}

= M^r^L^(coz^α)

= M_(cozα).

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Proof.

M(α) = \

= \

{M^I |I∈ P}

= M^r^L^(coz^α)

= M_(cozα).

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Proof.

M(α) = \

= \

{M^I |I∈ P}

= M^r^L^(coz^α)

= M_(cozα).

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Proof.

M(α) = \

= \

{M^I |I∈ P}

= M^r^L^(coz^α)

= M_(cozα).

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We say a ring is anOz-ringif every annihilator ideal of the ring is a principalz-ideal.

Proposition

A completely regular frameLis anOz-frameiffRLis anOz-ring.

Proof.

SupposeLis anOz-frame, and letQbe an annihilator ideal ofRL.

Then there is ana∈Lsuch thatQ=M_a^∗. SinceLis anOz-frame, a^∗ ∈CozL, henceQis a principalz-ideal, and thereforeRLis an Oz-ring.

Conversely, supposeRLis anOz-ring. For anya∈L,M_a^∗ is an annihilator ideal, and so, by hypothesis, there is ac ∈CozLsuch that M_a^∗=M_c, so thata^∗=c. ThereforeLis anOz-frame.

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Proposition

Proof.

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Proposition

Proof.

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The property of being anOz-ringis preserved by ring isomorphisms.

SinceX is anOz-spaceif and only ifOX is anOz-frame, and since C(X)∼=R(OX), we have the following result.

Corollary

A Tychonoff spaceX is anOz-spaceiffC(X)is anOz-ring. A realcompact space isOz iff its Stone- ˇCech compactification isOz.

Definition

A subspaceSof a topological spaceX isC-embeddedinX if every function inC(S)can be extended to a function inC(X).

An onto frame homomorphismh:L→Mis aC-quotientmap if for every frame homomorphismγ:OR→M there is a frame

homomorphismδ:OR→Lsuch thath◦δ=γ.

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Corollary

Definition

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Corollary

Definition

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Corollary

Definition

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In Lemma 2 of the article

B. Banaschewski,On the function ring functor in pointfree topology,Appl. Categ.

Structures13(2005), 305-328,

Banaschewski shows that a frame homomorphismh:L→Mis dense if and only if the ring homomorphismRh:RL→ RMis one-one.

Consequently,Rh:RL→ RM is an isomorphism if and only if

h:L→M is a denseC-quotientmap. This yields the following results.

Corollary

A frameLis anOz-frameiffλLis anOz-frameiffυLis anOz-frame. A Tychonoff spaceX is anOz-spaceiffυX is anOz-space.

Corollary

Every denseC-embeddedsubspace of anOz-spaceisOz. If a space has a denseC-embeddedsubspace which isOz, then the space itself isOz.

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Structures13(2005), 305-328,

Corollary

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Structures13(2005), 305-328,

Corollary

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Structures13(2005), 305-328,

Corollary

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Structures13(2005), 305-328,

Corollary

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ACKNOWLEDGEMENT

A

CKNOWLEDGEMENT

I wish to acknowledge financial assistance from the UNISA Topology and Category Theory Research Chair.

T

HANK YOU FOR YOUR ATTENTION

.

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ACKNOWLEDGEMENT

A

CKNOWLEDGEMENT

I wish to acknowledge financial assistance from the UNISA Topology and Category Theory Research Chair.

T

HANK YOU FOR YOUR ATTENTION