Wenfeng ZHANG Xiaoquan XU

1 Introduction

To generalize Dedekind’s pioneer construction of the real line by cuts of rational numbers(see[3]),MacNeille[12]introduced the famous normal completion for arbitrary posets.It is well-known that modularity and distributivity are not preserved under the formation of normal completions(see[2]).However,other notions of distributivity,such as Boolean algebras and Heyting algebras,are closed under normal completions(see[5–7,11,13,18–19]).In[4],Ern′e observed that continuity was not completion-invariant(see[5]),that is,the normal completion of a continuous domain is not always a continuous lattice.To obtain the completion-invariant property,Ern′e introduced a new concept of precontinuous posets by taking Frink ideals(see[8])instead of directed lower sets,and proved that a poset is precontinuous if and only if its normal completion is a continuous lattice.Generally speaking,continuity defined by the usual way is not completion-invariant.

As a common generalization of completely distributive lattices(see[14])and generalized continuous lattices(see[10])which were called quasicontinuous lattices in[9],Venugopalan introduced the concept of generalized completely distributive lattices which have many properties similar to those of completely distributive lattices(see[15]).In[10],Gierz and Lawson introduced the concept of a hypercontinuous lattice,which is also among the most successful generalizations of continuous lattices,to characterize a continuous lattice with the Hausdorf f interval topology.In[17],Yang and Xu proved that a complete lattice is hypercontinuous if and only if its order dual is generalized completely distributive.

In this paper,we generalize the concepts of hypercontinuous lattices and generalized completely distributive lattices to the setting of posets,and introduce the concepts of hypercontinuous posets and generalized completely continuous posets.It is proved that for a posetPthe following three conditions are equivalent:(1)Pis hypercontinuous;(2)the dual ofPis generalized completely continuous;(3)the normal completion ofPis a hypercontinuous lattice.So the hypercontinuity and generalized complete continuity are completion-invariant.Also in this paper,the relational representation and the intrinsic characterization of hypercontinuous posets are obtained.

For a posetP,letP(<ω)={FP:Fis finite}.The order dual ofPis written asPop.For allx∈P,AP,let↑x={y∈P:x≤y}and↑xand↓Aare defined dually.A↑andA↓denote the sets of all upper and lower bounds ofA,respectively.LetAδ=(A↑)↓andδ(P)={Aδ:AP}.(δ(P),)is called the normal completion,or the Dedekind-MacNeille completion ofP.By a completion-invariant property,we mean a property that holds for a posetPif and only if it holds for the normal completion ofP.For allU∈δ(P)andFδ(P),let↑δ(P){U}={V∈δ(P):UV}and↑δ(P)F={V∈δ(P):there existsU∈FwithUV};↓δ(P){U}and↓δ(P)Fare defined dually.

LetPbe a poset.The topology generated by the collection of setsP↓x(as a subbase)is called the upper topology and denoted byυ(P);the lower topologyω(P)onPis defined dually.The topologyθ(P)=υ(P)∨ω(P)is called the interval topology onP.Forx,y∈P,define a relationonPbyxyy∈intυ(P)↑x.

The following lemma is well-known(see[5]).

Lemma 1.1Let P be a poset.

(1)The maps(?)↑:(2P)op2P,AA↑and(?)↓:2P(2P)op,AA↓are order preserving.

(2)((?)↑,(?)↓)is a Galois connection between(2P)opand2P,that is,for all A,BP,B↑ABA↓.Thus both δ:2P2P,AAδ=(A↑)↓and δ?:2P2P,A(A↓)↑are closure operators.

(3)For all{Cj:j∈J}2P,

(4)Let L=δ(P).For all

Corollary 1.1Let P be a poset.Then the map eP:Pδ(P),x↓x is an order embedding of P in the normal completion δ(P)and

(1)ePpreserves all existing joins and meets;

(2)for all

Definition 1.1(see[16])A binary relation ρ on X is called finitely regular if for all(x,y)∈ρ,there exist u∈X and{v1,v2,···,vk}∈X(<ω)such that

(1)(u,y)∈ρ,and(x,vi)∈ρ for each i∈{1,2,···,k},and

(2)for all{s1,s2,···,sk}∈X(<ω)and t∈X,if(u,t)∈ρ,(si,vi)∈ρ for each i∈{1,2,···,k},then there exists j∈{1,2,···,k}such that(sj,t)∈ρ.

Definition 1.2(see[9–10])A complete lattice L is called hypercontinuous if and only if x=∨{y∈L:yx}for all x∈L.

Definition 1.3(see[15])For a complete lattice L,define a relationon2Lby AB if and only if for all SL,∨S∈↑B implies S∩↑AL is called a generalized completely distributive lattice if and only if↑x=∩{↑F:F∈P(<ω)and Fx}.

2 Hypercontinuous Posets

In this section,the concept of hypercontinuous posets is introduced,and we give intrinsic characterizations of hypercontinuous posets and show that a poset is hypercontinuous if and only if its normal completion is a hypercontinuous lattice.

Definition 2.1A poset P is called hypercontinuous if x∈{u∈P:ux}δfor all x∈P.Let i(x)={u∈P:ux}.

Remark 2.1IfX↓x,thenx=supXif and only ifx∈Xδ.Thus for complete lattices,the preceding definition of hypercontinuity is equivalent to Definition 1.2.

Theorem 2.1Let P be a poset.Then the following conditions are equivalent:

(1)P is hypercontinuous;

(2)If x,y∈P with xy,then there exist F∈P(<ω)and u∈P such that(i)x↓F,y↑u,and(ii)↓F∪↑u=P;

(3)The relationon P is finitely regular.

Proof(1)(2).Letx,y∈Pwithy.Then by(1),there existsu∈Pwithuxsuch thatuy.Thusx∈intυ(P)↑u.ChooseF∈P(<ω)such thatx∈P↓Fintυ(P)↑u↑u.ThenFandusatisfy the conditions of(i)and(ii)in(2).

(21).Suppose that there existsx∈Pwithxi(x)δ,and then there existsy∈Pwithi(x)↓ysuch thatxy.By(2),there existF∈P(<ω)andu∈Pthat satisfy the conditions(i)–(ii)in(2).Thenuxanduy,a contradiction toi(x)↓y.ThereforePis hypercontinuous.

(2)(3).Letx,y∈Pwithx?y.By(2),there existF={v1,v2,···,vk}∈P(<ω)andu∈Pthat satisfy the conditions(i)–(ii)in(2).For all{s1,s2,···,sk}∈P(<ω)andt∈P,ifutandsivifor eachi∈{1,2,···,k},then there existsj∈{1,2,···,k}such thatt≤vj.Thussjtsincesjvj.

(3)(2).Letx,y∈Pwithxy.By(3),there existF={v1,v2,···,vk}∈P(<ω)andu∈Psuch that 1?uy,x↓F,and 2?for all{s1,s2,···,sk}∈P(<ω)andt∈P,ifut,sivifor eachi∈{1,2,···,k},then there existsj∈{1,2,···,k}such thatsjt.

For allz∈P,lett=z,andsi=zfor alli∈{1,2,···,k}.By 2?,we haveu≤t=zor there existsj∈{1,2,···,k}such thatz=sj≤vj,i.e.,↑u∪↓F=P.

Corollary 2.1Let P be a hypercontinuous poset.Then(P,θ(P))is T2.

Theorem 2.2For a poset P,the following two conditions are equivalent:

(1)P is hypercontinuous;

(2)(δ(P),)is a hypercontinuous lattice.

Proof(1)(2).IfAδ,Bδ∈δ(P)withAδBδ,thenABδ.Thus there existsx∈AwithxBδ.Hence there existsy∈PwithB↓ysuch thatxy.By Theorem 2.1,there existF∈P(<ω)andu∈Psuch that(i)x↓F,y↑u,and(ii)↓F∪↑u=P.

LetF={↓a:a∈F}.ThenF∈δ(P)(<ω).We show thatFand↓usatisfy the conditions(i)–(ii)in(2)of Theorem 2.1.Firstly,ifAδ∈↓δ(P)F,then there existsa∈FwithAδ↓a.Thusx∈AAδ↓a,a contradiction tox↓F;ifBδ∈↑δ(P){↓u},thenu∈↓uBδ↓y,a contradiction toy↑u.Then we show that↓δ(P)F∪↑δ(P){↓u}=δ(P).For allCδ∈δ(P),if↓uCδ,i.e.,uCδ,then there existsm∈PwithC↓msuch thatum.Thusm∈↓F.Then there existsa∈Fsuch thatm≤a.Therefore,Cδ↓m↓a,which impliesCδ∈↓δ(P)F.

(2)?(1).Ifx,y∈Pwithxy,then↓x↓y.By Theorem 2.1,there existF={,,···,}∈δ(P)(<ω)andBδ∈δ(P)such that(i)↓xδ(P)F,↓y?↑δ(P){Bδ},and(ii)↓δ(P)F∪↑δ(P){Bδ}=δ(P).

Since↓x↓δ(P)F,xfor alli∈{1,2,···,k}.Thus there existsmi∈PwithAi↓misuch thatxmi.LetF={m1,m2,···,mk}.Since↓y↑δ(P){Bδ},there existsu∈Bδwithuy.Then we show thatFandusatisfy the conditions of(i)–(ii)in(2)of Theorem 2.1.Obviously,x↓Fandy↑u.We show that↓F∪↑u=P.For allz∈P,ifuz,thenBδ↓zsinceu∈Bδ.Thus there existsi∈{1,2,···,k}withz∈↓z↓mi.Soz∈↓F.Therefore,↓F∪↑u=P.

Definition 2.2(see[5])A map f between posets P and Q is said to be cut-stable if f[A↑]↓=f[A]↑↓and f[A↓]↑=f[A]↓↑for all AP.

Proposition 2.1(see[5])A map f from a poset P into a complete lattice L is cut-stable if and only if there exists a(unique)complete homomorphism g from δ(P)into L such that f=g?eP.

A subcategory A of a category C is called a reflective subcategory(see[1])of C,if for each C-objectC,there exists an A-objectA0and a C-morphismr:C→A0such that for each A-objectAand C-morphismf:C→Athere exists a unique A-morphismg:A0→Asuch thatf=g?r.

By Proposition 2.1 and Theorem 2.2,we immediately have the following theorem.

Theorem 2.3The category of hypercontinuous lattices with complete homomorphisms is a full reflective subcategory of the category of hypercontinuous posets with cut-stable maps.

3 Generalized Completely Continuous Posets

In this section,the concept of generalized completely continuous posets is introduced,and we give intrinsic characterizations of generalized completely continuous posets and show that a poset is generalized completely continuous if and only if its normal completion is a generalized completely distributive lattice if and only if its order dual is a hypercontinuous poset.

Definition 3.1Let P be a poset,x∈P,A,BP.We say that:

(1)A is completely way below B,in symbols AB if for all SP↑B∩Sδimplies↑A∩SWe write F{x}for Fx.Let(x)={F∈P(<ω):Fx}.

(2)P is generalized completely continuous if for all x∈P,↑x=∩{↑F:F∈(x)}.

Remark 3.1For complete lattices,the preceding definition of generalized completely continuity is equivalent to Definition 1.3.

Proposition 3.1For a poset P,x∈P,AP,the following conditions are equivalent:

(1)Ax;

(2)x(P↑A)δ;

(3)x∈P(P↑A)δ↑A.

Proof(1)?(2).Ifx∈(P↑A)δ,then by the definition of,↑A∩(P↑A)?,which is impossible.

(2)?(1).If there existsS?Pwithx∈Sδsuch that↑A∩S=?,thenx∈Sδ?(P↑A)δ,a contradiction to(2).

(2)?(3).Obviously.

Proposition 3.2For a poset P,A?P,the following two conditions are equivalent:

(1)AA;

(2)↑A=P(P↑A)δ.

Proof(1)?(2).Obviously,P(P↑A)δ?↑Aby Lemma 1.1.If there existsy∈↑AbutyP(P↑A)δ,then by(1),↑A∩(P↑A)?,which is impossible.

(2)?(1).If there existsS?Pwith↑A∩Sδ?such that↑A∩S=?,thenSδ?(P↑A)δ.By(2),we have↑A?PSδ,a contradiction to↑A∩Sδ?.

Theorem 3.1For a poset P,the following two conditions are equivalent:

(1)P is generalized completely continuous;

(2)(δ(P),?)is a generalized completely distributive lattice.

Proof(1)?(2).We show that↑δ(P){Aδ}=∩{↑δ(P)F:F∈δ(P)(<ω),Ffor allAδ∈δ(P).Obviously,↑δ(P){Aδ}?∩{↑δ(P)F:F∈δ(P)(<ω),FWe show that∩{↑δ(P)F:F∈δ(P)(<ω),F?↑δ(P){Aδ}.IfAδBδ,thenABδ.Thus there existsx∈AwithxBδ.Hence there existsy∈PwithB?↓ysuch thatxy.By(1),there existsF∈P(<ω)withFxsuch thaty↑F.LetF={↓u:u∈F}.ThenF∈δ(P)(<ω).We show thatFAδandBδ↑δ(P)F.IfBδ∈↑δ(P)F,then there existsu∈Fwith↓u?Bδ.Thusu∈Bδ?↓y,a contradiction toy↑F.Then we show thatFAδ.For all{:i∈I}?δ(P)withwe haveby Lemma 1.1.Sincex∈A?AδandFx,we haveThus there existu∈Fandi∈Iwithu∈Sinceis a lower set,↓u?Hence↑δ(P)F∩{:i∈I}?.Therefore,δ(P)is a generalized completely distributive lattice.

(2)?(1).Obviously,↑x?∩{↑F:F∈?(x)}for allx∈P.Ifxy,then↓x↓y.By(2),there existsF={A1,A2,···,Ak}∈δ(P)(<ω)withF↓xsuch that↓y↑δ(P)F,i.e.,Ai↓yfor alli∈{1,2,···,k}.Thus there existsyi∈Aiwithyiyfor alli∈{1,2,···,k}.LetF={y1,y2,···,yk}.Obviously,y↑F.We show thatFxfor allS?Pwithx∈Sδ.SinceSδis a lower set,↓x?SinceF↓x,↑δ(P)F∩{↓s:s∈S}?.Thus there existi∈{1,2,···,k}ands∈Ssuch thatyi∈Ai?↓s.So↑F∩S?.HenceFx.Sincey↑F,y∩{↑F:F∈?(x)}.Therefore,↑x=∩{↑F:F∈?(x)}.

Theorem 3.2(see[17])A complete lattice L is a hypercontinuous lattice if and only if Lopis a generalized completely distributive lattice.

By Theorems 2.2,3.1–3.2,we obtain the following result.

Corollary 3.1A poset P is hypercontinuous if and only if Popis generalized completely continuous.

By Theorem 2.1 and Corollary 3.1,we obtain the following result.

Corollary 3.2For a poset P,the following conditions are equivalent:

(1)P is generalized completely continuous;

(2)for all x,y∈P with xy,there exist u∈P and finite set F∈P(<ω)such that(i)x↓u,y↑F,and(ii)↓u∪↑F=P;

(3)on P is finitely regular.

Corollary 3.3Let P be a generalized completely continuous poset.Then(P,θ(P))is T2.

Corollary 3.4The category of generalized completely distributive lattices with complete homomorphisms is a full reflective subcategory of the category of generalized completely continuous posets with cut-stable maps.

AcknowledgementThe authors would like to thank the referees for their very careful reading of the manuscript and valuable comments.

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