January 4, 2012 2:26 Continuity CONTINUITY IN INFORMATION ALGEBRAS: A SURVEY ON 2 1 THE RELATIONSHIP BETWEEN TWO TYPES OF 0 INFORMATION ALGEBRAS 2 n a J XUECHONGGUAN∗ 2 College of Mathematics and Information Science Shaanxi Normal University ] Xi’an, 710062, China I A [email protected] . s c YONGMINGLI [ College of Computer Science Shaanxi Normal University 1 Xi’an, 710062, China v 4 [email protected] 1 4 0 Received24June2010 . Revised(reviseddate) 1 0 2 Inthispaper, the continuity andstrongcontinuity indomain-freeinformationalgebras 1 andlabeledinformationalgebrasareintroducedrespectively.Amoregeneralconceptof : continuous function which is defined between two domain-free continuous information v algebrasispresented.Itisshownthat,withtheoperationscombinationandfocusing,the i setofallcontinuousfunctionsbetweentwodomain-frees-continuousinformationalgebras X forms a new s-continuous information algebra. By studying the relationship between r domain-free information algebras and labeled information algebras, it is demonstrated a thattheydocorrespondtoeachother ons-compactness. Keywords: domain-freecontinuous informationalgebra; labeledcontinuous information algebra;continuous function;compactness. 1. Introduction Inferenceunderuncertaintyisacommonproblemintherealworld.Thus,forpieces of information from different sources, there always exist two fundamental aspects that to combine information and to exact information on a designated domain. Based on the above consideration, the valuation-based system (VBS) was first in- 1 troduced by Shenoy. Kohlas, in Ref. 2, has exactly introduced the concept of information algebra. We can see that information algebra is an algebraic structure links up with local computation and inference for treating uncertainty or, more ∗Work Address: College of Mathematic Science, Xuzhou Normal University, Xuzhou, 221116, China. 1 January 4, 2012 2:26 Continuity 2 X.C. Guan and Y.M. Li generally, information and knowledge. It gives a basic mathematical model for de- 2 3 4 scribingthe modes ofinformationprocessing.Recentstudies showedthat the framework of information algebra covers many instances from constraint systems, Bayesiannetworks,Dempster-Shaferbelieffunctionstorelationalalgebra,logicand etc. In view of the feasibility of computer processing information, Kohlas gave the notions of domain-free compact information algebra and labeled compact informa- tion algebra successively in the study of representation of information algebras. In the light of the previous conclusions, we know that there exists a correspondence between domain-free information algebra and labeled information algebra, that is, from a domain-free information algebra, we can construct its associated labeled information algebra, and vice versa. But, labeled compact information algebras introducedinRef.3donotnecessarilyleadtodomain-freecompactinformational- 3 gebras,aswehaveseenintheexampleofcofinitesets. Itnaturallyraisesaquestion thatwhetherwecanpresentanimproveddefinitionoflabeledcompactinformation algebra such that its associated domain-free information algebra is compact and strong compact respectively. Accordingly, in this paper, we redefine the notions of labeledcontinuous informationalgebraanddomain-freecontinuous informational- gebrarespectively.Obviouslycompactinformationalgebrasdefinedin the previous literature canbe seen as a special caseof continuous informationalgebras.As a re- sult,theconclusionswhichareobtainedincontinuousinformationalgebrasarealso more extensive. It should be noted that the definitions in this paper are different fromcontinuousinformationalgebrapresentedinRef.3.Themaindifferenceexists in the characterizationof continuity in labeled information algebras. The main work of this paper is as follows. Basing on the notions of continu- ous information algebras, we discuss the correspondence of continuity in labeled information algebras and domain-free information algebras. The accordance on s- compactnessininformationalgebrasthusfollowsfromtheconclusionsoncontinuity, that is, a labeled s-compactinformation algebra induces its associateddomain-free s-compact information algebra and, in turn, a domain-free s-compact information algebra induces its associated labeled s-compact information algebra too. We also present the equivalent statements of some definitions introduced in the paper, and especially study the properties of function spaces of domain-free continuous infor- mation algebras. Thepaperisorganizedasbelow.Section2brieflyreviewssomebasicnotionson information algebra.In Section 3 we introduce the concept of domain-free continu- ous information algebra and discuss the property of continuous function spaces. In Section 4 we give the notion of labeled continuous information algebra and focus on the relationship on continuity and compactness in labeled information algebras and domain-free information algebras. January 4, 2012 2:26 Continuity Continuity inInformation Algebras 3 2. Preliminaries Let’srecallsomedefinitionsandnotationsinthetheoryofinformationalgebra.For afullintroduction,wecanrefertoRef.2-6.Inthisstudy,thefundamentalelements of an information algebra are valuations. In general, a valuation is a function that provides possible elements of a field for variables.Here a valuationrepresents some knowledge and information which may be a function, tuple or symbol. Let Φ be a set of valuations, and let D be a lattice. Suppose there are three operations defined: 1.Labeling: Φ → D;φ 7→ d(φ), where d(φ) is called the domain of φ. Let Φ s denote the set of all valuations with domain s. 2.Combination: Φ×Φ→Φ;(φ,ψ)7→φ⊗ψ, 3.Marginalization:Φ×D →Φ;(φ,x)7→φ↓x, for x≤d(φ). If the system (Φ,D) satisfies the following axioms, it is called a labeled infor- mation algebra: 1.Semigroup:Φisassociativeandcommutativeundercombination.Foralls∈D thereisanelemente withd(e )=ssuchthatforallφ∈Φwithd(φ)=s,e ⊗φ= s s s φ. Here e is called a neutral element of Φ . s s 2. Labeling: For φ,ψ ∈Φ, d(φ⊗ψ)=d(φ)∨d(ψ). 3. Marginalization:For φ∈Φ,x∈D,x≤d(φ),d(φ↓x)=x. 4. Transitivity: For φ∈Φ and x≤y ≤d(φ),(φ↓y)↓x =φ↓x 5. Combination: For φ,ψ ∈Φ with d(φ)=x,d(ψ)=y,(φ⊗ψ)↓x =φ⊗ψ↓x∧y. 6. Stability: For x,y ∈D,x≤y, e↓x =e . y x 7. Idempotency: For φ∈Φ and x∈D,x≤d(φ), φ⊗φ↓x =φ. The items putting forward in the above definition are the axiomatic presenta- tions of some basic and logical principles in the process of handling information. In fact, the algebraic structure shown in the definition covers many instances from expert systems, constraint systems and possibility theory to relational algebra and logic. We now look at some examples of labeled information algebras. 7 Example 1. (ConstraintSystem)Aconstraintsystem isatupleCS =hS,D,Vi, wherehS,+,×,0,1iisasemiring,V isatotallyorderedsetofvariablesviaordering ≺. D is a finite set which contains at least two elements, called the domain of variables. A tuple hS,+,×,0,1i is defined to be a semiring, if it satisfies that both operations+and×arecommutativeandassociative,and×distributesover+.The element0 isa unitelementof+ andanabsorbingelementof×,1is a unitelement 4 of ×. A semiring S is called c-semiring, if it is such that for all a∈S,a+1=1. A tuple c=hdef,coni is called a constraint over CS, where (i) con⊆V, it is called the type of the constraint, denoted by d(c)=con; (ii) def :D|con| →S, where |con| is the cardinality of con. Let C denote the set of all constraints over CS. Two operations are defined as follows: 1.Combination⊗:Fortwoconstraintsc =hdef ,con i,c =hdef ,con i,their 1 1 1 2 2 2 combination, written c ⊗c , is the constraint hdef,coni with con = con ∪con 1 2 1 2 January 4, 2012 2:26 Continuity 4 X.C. Guan and Y.M. Li and def :D|con| →S is defined as: def(x)=def1(x↓con1)×def2(x↓con2),∀x∈D|con|, where x↓con1, called tuple projection, is defined as follows: Suppose x = hx ,···,x i ∈ D|con|, con = {v′,···,v′ } ⊆ con = {v ,···,v }, where v ≺ v 1 k 1 1 m 1 k i j and vi′ ≺vj′ if i<j, then x↓con1 =ht′1,···,t′mi, where t′i =xj if vi′ =vj. 2. Projection ⇓: For a constraint c = hdef,coni, if I ⊆ con, the projection of c over I, written c⇓I, is the constraint hdef′,Ii with def′(x)= def(z),∀x∈D|I|. z∈D|coPn|:z↓I=x With the three operations combination, projection and type, the system (C,P(V)) induced by a semiring S is a labeled information algebra if, and only if, the semiring S is such that a×(a+b)=a for all a,b∈S. In fact, if S satisfies with a × (a + b) = a for all a,b ∈ S, then S is a c- 8 semiring with the idempotent operation ×. Hence (C,P(V)) is an information 4 algebra. Conversely, let (C,P(V)) be an information algebra, a variable v ∈ V, and D = {y ,y ,···,y }(n ≥ 2). If a,b ∈ S, we take a constraint c = hdef,coni, 1 2 n where con = {v} and def : D → S is defined as def(y ) = a,def(y ) = b and 1 2 def(y)=0forallothery ∈D.Bytheidempotencyofinformationalgebra,wehave c⊗c⇓∅ =c, then def(y )=def(y )× def(y), i.e., a×(a+b)=a. 1 1 yP∈D Example 2. (softset)LetU beaninitialuniversesetandletE beasetofparam- eters which usually are initial attributes, characteristics,or properties of objects in U. Let P(U) denote the power set of U and A ⊆ E. A pair (F,A) is called a soft 9 set over U, where F is a mapping given by F :A→P(U). A soft set (F,A) over U is said to be a null soft set, if for all e∈A,F(e)=∅. There are three operations defined: 1. Labeling d: For a soft set (F,A), we define d((F,A)) =A. 2. Projection ↓ 10: If B ⊆ A, we define (F,A)↓B to be a soft set (G,B) such that for all b∈B, G(b)=F(b). 11 3. Extended intersection⊓ : The extended intersectionof two soft sets (F,A) and(G,B)overacommonuniverseU is the softset(H,C), whereC =A∪B, and ∀e∈C, F(e), if e∈A−B; H(e)=G(e), if e∈B−A;  F(e)∩G(e), if e∈A∩B.  We write (F,A)⊓(G,B)=(H,C). We are going to show (F,P(E)) is an information algebra with the three oper- ations d,↓ and ⊓ defined as above, where F is the set of all soft sets over U. 1.Semigroup:F isassociativeandcommutativeunder⊓.Thenullsoftset(∅,∅) is a neutral element such that (F,A)⊓(∅,∅)=(F,A) for all soft set (F,A). January 4, 2012 2:26 Continuity Continuity inInformation Algebras 5 2. The axioms of labeling, marginalization, transitivity and idempotency are clear. 3. Combination: For (F,A),(G,B) ∈ F, if A ⊆ S ⊆ A∪B, we need to show ((F,A)⊓(G,B))↓S =(F,A)⊓(G,B)↓S∩B. Infact, let (F,A)⊓(G,B)=(H,A∪B) and (F,A)⊓(G,B)↓S∩B =(H′,S). For all e∈S, we have F(e), if e∈S∩(A−B); ′ H(e)=H (e)=G(e), if e∈S∩(B−A);  F(e)∩G(e), if e∈A∩B.  Then ((F,A)⊓(G,B))↓S =(F,A)⊓(G,B)↓S∩B. Hence (F,P(E)) is a labeled information algebra. Nowwelookatthe notionofdomain-freeinformationalgebra.Asystem(Ψ,D) with two operations defined, 1. Combination: Ψ×Ψ→Ψ;(φ,ψ)7→φ⊗ψ, 2. Marginalization:Ψ×D →Ψ;(ψ,x)7→ψ⇒x, We impose the following axioms on Ψ and D, and it is called a domain-free information algebra: 1. Semigroup: Ψ is associative and commutative under combination, and there is an element e such that for all ψ ∈Ψ with e⊗ψ =ψ⊗e=ψ. 2. Transitivity: For ψ ∈Φ and x,y ∈D,(ψ⇒y)⇒x =ψ↓x∧y. 3. Combination: For φ,ψ ∈Ψ,x∈D, (φ⇒x⊗ψ)⇒x =φ⇒x⊗ψ⇒x. 4. Support: For ψ ∈Ψ, there is an x∈D such that ψ⇒x =ψ. 5. Idempotency: For ψ ∈Ψ and x∈D,ψ⊗ψ⇒x =ψ. For simplicity, information algebra is used as a general name for labeled infor- mation algebras and domain-free information algebras. While, we can judge that whether an information algebra is “labeled” or not from the context. An abstract example of domain-free information algebra is given below. Example 3. Let Φ=[0,1], D ={0,1}. We define the following operations: Combination: ∀φ,ψ ∈Φ,φ⊗ψ =max{φ,ψ}; Focusing: ∀φ∈Φ, (i)φ⇒1 =φ. φ, if φ∈[0,1]; (ii) φ⇒0 = 2 (cid:26)1, if φ∈[1,1]. 2 2 We have φ≪ψ if, and only if, φ<ψ or φ=ψ =0. It’s clear that the element 0 is a neutral element and the axioms of transitivity andsupportarecorrect.Since φ⇒x ≤φfor all φ∈Φ,x∈D,we haveφ⊗φ⇒x =φ, thus the axiom of idempotency holds. Now,inorderto provethat(Φ,D)isaninformationalgebra,itsufficestocheck the axiom of combination, that is, for φ,ψ ∈Φ and x∈D, (φ⇒x⊗ψ)⇒x =φ⇒x⊗ψ⇒x. January 4, 2012 2:26 Continuity 6 X.C. Guan and Y.M. Li By the definition of focusing, it only need to show the case of x=0. (i) If φ∈[1,1], then (φ⇒0⊗ψ)⇒0 = 1 =φ⇒0⊗ψ⇒0. 2 2 (ii) If φ∈[0,1],ψ ∈[0,1], then (φ⇒0⊗ψ)⇒0 =φ⊗ψ =φ⇒0⊗ψ⇒0. 2 2 (iii) If φ∈[0,1],ψ ∈[1,1], then (φ⇒0⊗ψ)⇒0 = 1 =φ⇒0⊗ψ⇒0. 2 2 2 Insummary,wehave(φ⇒x⊗ψ)⇒x =φ⇒x⊗ψ⇒x.Then(Φ,D)isadomain-free information algebra. If(Φ,D)isaninformationalgebra,wewriteψ ≤φ,meansaninformationφ∈Φ is more informative than another information ψ ∈ Φ, i.e., ψ ⊗φ = φ. The order relation ≤ is a partial order on an information algebra (Φ,D). In this paper, the order relation ≤ induced by the operation of combination is a default order on an information algebra. In Ref. 2, Kohlas gave a specific method to realize the transform between domain-free information algebra and labeled information algebra as follows. In a labeled information algebra (Φ,D), we define for φ∈Φ and y ≥d(φ), φ↑y =φ⊗e . y φ↑y is called the vacuous extension of φ to the domain y. Now we consider a con- gruence relation σ: φ≡ψ(mod σ) if, and only if φ↑x∨y =ψ↑x∨y, where x = d(φ),y = d(ψ). In the (Φ/σ,D), the two operations, combination and focusing, are defined as follows: Combination: [φ] ⊗[ψ] =[φ⊗ψ] ; σ σ σ Focusing: [φ]⇒x =[(φ↑x∨d(φ))↓x] . σ σ Then (Φ/σ,D) is a domain-free information algebra, and we say (Φ/σ,D) is the associated domain-free information algebra with (Φ,D). Conversely, if (Φ,D) is a domain-free information algebra, let Ψ={(φ,x):φ∈Φ,φ=φ⇒x}. The three operations are defined on Ψ as follows: 1. Labeling: For (φ,x)∈Ψ define d(φ,x)=x; 2. Combination: For (φ,x),(ψ,y)∈Ψ define (φ,x)⊗(ψ,y)=(φ⊗ψ,x∨y); 3. Marginalization:For (φ,x)∈Ψ and y ≤x define (φ,x)↓y =(φ⇒y,y). Then (Ψ,D) is a labeled information algebra, and is called the associated labeled information algebra with (Φ,D). Attheendofthissection,wegivesomebasicnotionsinlatticetheory.Let(L,≤) beapartiallyorderedset.AiscalledadirectedsubsetofL,ifforalla,b∈A,there is a c ∈ A such that a,b ≤ c. We write ∨A for the least upper bound of A in L if it exists. L is called a sup-semilattice, if a∨b exists for all a,b ∈ L. If every January 4, 2012 2:26 Continuity Continuity inInformation Algebras 7 subset A⊆L has a greatest lower bound or a least upper bound in L, we say L is a complete lattice. 12 Lemma 1. Let L be a sup-semilattice with bottom element 0. Then L is a complete lattice if, only if every directed subset A ⊆ L has the least upper bound ∨A. 12 Definition 1. LetLbeapartiallyorderedset.Fora,b∈Lwewritea≪b,and saya way-belowb if, for any directedset X ⊆L,fromb≤∨X it follows thatthere is a c∈X such that a≤c. We call a∈L a finite (compact) element, if a≪a. 13 Definition 2. For a complete lattice L, if for all a ∈ L, a = ∨{b : b ≪ a}, we call L a continuous lattice. Moreover,if for all a∈L,a=∨{b:b≪b≤a}, we call L an algebraic lattice. 3. Domain-free Continuous Information Algebras Inthispartwewillgivetheconceptofdomain-freecontinuousinformationalgebra, and discuss the properties of continuous function spaces of domain-free continuous informationalgebras.Informationalgebrasmentionedinthissectionarealldomain- free information algebras. 3.1. Definitions Firstly, we give the following lemma which contains some simple and important results about the partially ordered relation induced by the operation combination in information algebras. 2 Lemma 2. If (Φ,D) is an information algebra, then 1. φ⇒x ≤φ. 2. φ⊗ψ =sup{φ,ψ}. 3. φ≤ψ implies φ⇒x ≤ψ⇒x. 4. x≤y implies φ⇒x ≤φ⇒y. Ingeneral,only“finite”informationcanbe treatedincomputers.Therefore,for example in domain theory, a structure that each information can be approximated by these “finite” information has been proposed. The concept of compact informa- 2 tionalgebraintroducedby Kohlas stems fromthe ideaabove.Ofcourse,the idea ofapproximationalsopromptustoconsideratypeofinformationalgebrathateach element φ in this system could be approximated with pieces of information which is “relatively finite” or “way-below”φ. Then the notion of continuous information algebra is proposed next. Definition 3. Asystem(Φ,Γ,D),where(Φ,D)is adomain-freeinformationalge- bra,thelatticeDhasatopelement,Γ⊆Φisclosedundercombinationandcontains January 4, 2012 2:26 Continuity 8 X.C. Guan and Y.M. Li the empty information e, satisfying the following axioms of convergence and den- sity (resp. strong density), is called a domain-free continuous (resp. s-continuous) information algebra. Γ is called a basis for the system (Φ,D). 1. Convergency:If X ⊆Γ is a directed set, then the supremum ∨X exists. 2. Density(D1): For all φ∈Φ, φ=∨{ψ ∈Γ:ψ ≪φ}. 3. Strong density(SD1): For all φ∈Φ and x∈D, φ⇒x =∨{ψ ∈Γ:ψ =ψ⇒x ≪φ}. Moreover, if a domain-free continuous (resp. s-continuous) information algebra (Φ,Γ,D) satisfies the axiom of compactness, then we call (Φ,Γ,D) a domain-free compact (resp. s-compact) information algebra. 4. Compactness: If X ⊆ Γ is a directed set, and φ ∈ Γ such that φ ≤ ∨X then there exists a ψ ∈X such that φ≤ψ. Sometime, for simplicity of expression, we directly say an information algebra is continuous or it has continuity if it is a continuous information algebra. For the other concepts in the definition, there exist some similar forms of address. 3 Lemma 3. Let (Φ,Γ,D) be a compact information algebra, then the following holds: 1. ψ ∈Γ if, and only if ψ ≪ψ. 2. If ψ ∈Γ, then ψ ≪φ if, and only if ψ ≤φ. Foraninformationalgebra(Φ,D),wedenotethesetofallthefinite elementsof Φ by Φ , i.e.,Φ ={φ∈Φ:φ≪φ}. If(Φ,Γ,D) is a compactinformationalgebra, f f byLemma3,wehaveΓ=Φ .Therefore,wealwaysdenotea(domain-free)compact f information algebra by (Φ,Φ ,D). In addition, by Lemma 3 and the definition of f way-belowrelation,wealsocannaturallygetthat,ans-continuous(resp.continuous) information algebra (Φ,D) is s-compact(resp. compact) if, and only if, the set Φ f is a basis for the s-continuous(resp. continuous) information algebra (Φ,D). A simple example of s-continuous information algebra but not s-compact infor- mation algebra is presented as follows. Example 4. Let an information algebra (Φ,D) with the operations combination and focusing be defined as in Example 3. We are going to show that the equation φ⇒x = ∨{ψ ∈ Φ : ψ = ψ⇒x ≪ φ} is true for all φ ∈ Φ and x ∈ D. In fact, for the case of x = 0, if φ ∈ [0,1], then 2 ∨{ψ ∈[0,1]: ψ ≪φ} =φ =φ⇒0; otherwise, if φ∈ [1,1], then ∨{ψ ∈ [0,1] : ψ ≪ 2 2 2 φ} = 1 = φ⇒0. Thus φ⇒0 = ∨{ψ ∈ [0,1] : ψ ≪ φ} = ∨{ψ ∈ Φ : ψ = ψ⇒0 ≪ φ} 2 2 for all φ ∈ Φ. For the case of x = 1, the equation is clearly true. Hence (Φ,D) is s-continuous and Φ is a basis. But it is not s-compact, because Φ ={0}. f Proposition 1. If (Φ,Γ,D) is an s-continuous information algebra, then {ψ ∈Γ: ψ⇒x =ψ ≪φ} is directed for all φ∈Φ and x∈D. January 4, 2012 2:26 Continuity Continuity inInformation Algebras 9 Similarly,if(Φ,Γ,D)isacontinuousinformationalgebra,then{ψ∈Γ:ψ ≪φ} is directed for all φ∈Φ. Proof. We only prove the case of an s-continuous information algebras. Let ψ ,ψ ∈{ψ ∈Γ : ψ⇒x =ψ ≪φ}. Since Γ is closed under combination, we 1 2 have η ∈Γ if η =ψ ⊗ψ . By the monotonicity of the operation focusing, we have 1 2 ψ =ψ⇒x ≤ η⇒x(i = 1,2). So η ≤ψ⇒x ≤ η, that is, η =η⇒x. Next we show that i i η ≪ φ. If X ⊆ Φ is directed and φ ≤ ∨X, by ψ ,ψ ≪ φ, there exist φ ,φ ∈ X 1 2 1 2 such that ψ ≤ φ and ψ ≤ φ . Since X is directed, there is a φ ∈ X such that 1 1 2 2 3 φ ,φ ≤φ .Thenψ ,ψ ≤φ .We obtainthatη ≤φ andthusη ≪φ.Bywhatwe 1 2 3 1 2 3 3 have proved, we have η ∈{ψ ∈Γ:ψ =ψ⇒x ≪φ}. Hence {ψ ∈Γ:ψ =ψ⇒x ≪φ} is a directed set. Proposition 2. Let (Φ,D) be an information algebra. The lattice D has a top element. Then (Φ,D) is continuous (resp. s-continuous) if, and only if there exists a set Υ ⊆ Φ such that Υ contains the empty information e and satisfies the two conditions of convergency and density(D2) (resp. strong density(SD2)): 1. Convergency: If X ⊆Υ is a directed set, then the supremum ∨X exists. 2. Density(D2): For all φ∈Φ, {ψ ∈Υ:ψ ≪φ} is directed and φ=∨{ψ ∈Υ:ψ ≪φ}. 3. Strong density(SD2): For all φ ∈ Φ and x ∈ D, {ψ ∈ Υ : ψ = ψ⇒x ≪ φ} is directed and φ⇒x =∨{ψ ∈Υ:ψ =ψ⇒x ≪φ}. Proof. By Proposition 1, the necessary condition is clear. Conversely,assume that Υ is a set which satisfies the assumptions. We need to find a basis Γ for (Φ,D). We show that (Φ,≤) is a complete lattice if Υ satisfies the two conditions of convergency and density(D2). Firstly, {ψ ∈ Υ : ψ ≪ φ,φ ∈ X} is also directed if X ⊆ Φ is directed. Assume that ψ ,ψ ∈ {ψ ∈ Υ : ψ ≪ φ,φ ∈ X}, then there 1 2 exist φ ,φ ∈ X such that ψ ≪ φ ,ψ ≪ φ . So there is a φ ∈ X such that 1 2 1 1 2 2 ψ ,ψ ≪ φ because X is directed. Since φ = ∨{ψ ∈ Υ : ψ ≪ φ}, by the definition 1 2 of way-below relation and the directness of set {ψ ∈ Υ : ψ ≪ φ}, there exists a η ∈ Υ such that ψ ,ψ ≤ η ≪φ. This proves that {ψ ∈Υ : ψ ≪φ,φ ∈X} is also 1 2 directed.Bytheaxiomofconvergency,∨{ψ ∈Υ:ψ ≪φ,φ∈X}exists.Obviously, ∨X = ∨{ψ ∈ Υ : ψ ≪ φ,φ ∈ X}. By Lemma 2(2), (Φ,D) is a sup-semilattice. Since empty information e ∈ Φ is the bottom element, we obtain that (Φ,≤) is a complete lattice by Lemma 1. Clearly the condition of strong density(SD2) is strongerthan density(D2), then (Φ,≤) is also a complete lattice if Υ satisfies the convergency and strong den- sity(SD2). January 4, 2012 2:26 Continuity 10 X.C. Guan and Y.M. Li Now we claim that Γ = Φ which is closed under combination is a basis for the system. Firstly, since (Φ,≤) is a complete lattice, the condition of convergency of Γ holds, and ∨{ψ ∈Γ:ψ ≪φ}, ∨{ψ∈Γ:ψ =ψ⇒x ≪φ} exist. If the axiom of density(D2) holds, then φ = ∨{ψ ∈ Υ : ψ ≪ φ} ≤ ∨{ψ ∈ Γ : ψ ≪ φ} ≤ φ. We conclude that ∨{ψ ∈ Γ : ψ ≪ φ} = φ. Hence Γ is a basis for the continuous information algebra (Φ,D). Similarly, if the axiom of strong density(SD2) holds, then φ⇒x = ∨{ψ ∈ Υ : ψ = ψ⇒x ≪ φ} ≤ ∨{ψ ∈ Γ : ψ = ψ⇒x ≪ φ} ≤ φ⇒x. We obtain that ∨{ψ ∈ Γ : ψ =ψ⇒x ≪φ}=φ⇒x. Hence Γ is a basis for the s-continuous informationalgebra (Φ,D). By Proposition2, we givean equivalentdefinition of continuous informational- gebra.AndweknowthatthesubsetofabasisΓconsistingofalltheelementswhich approximate to an element φ in a continuous information algebra is directed. It is convenient for discussing the problem about continuous functions which is defined in the next subsection. But, it is worth noting that the set Υ in the Proposition2, satisfying the convergency and density(D2), is not necessarily closed under combi- nation, although we can find out a basis Γ which is closed under combination for the system. An simple example rooted in lattice theory is shown as follows. Example 5. Let X be an infinite set, Φ = P(X) and D = {1}. We define the operations combination and focusing as follows: Combination: A⊗B =A∪B,∀A,B ∈Φ; Focusing: A⇒1 =A,∀A∈Φ. The system (Φ,D) is an information algebra. Let Γ be the set of all the finite subsets of X. It is clear that C ≪ C if C ∈ Γ. For each set A ⊆ X, it can be representedby the combinationof the directed family of allfinite subsets ofA, i.e., A = ∨{B ∈ Γ : B ≪ A} and {B ∈ Γ : B ≪ A} is directed. Suppose that Y ⊂ X is an infinite proper subset. We write Γ∗ = Γ ∪{Y}. Obviously, for all A ∈ Φ, A = ∨{B ∈ Γ∗ : B ≪ A} and {B ∈ Γ∗ : B ≪ A} is still directed. While Γ∗ is not closedundercombination.Forexample,wehave{x},Y ∈Γ∗ wherex∈X−Y,but {x}⊗Y 6∈Γ∗. According to the proof in Proposition 2, the following theorem can be drawn immediately. It gives a compact expression for these notions defined in Definition 3. Theorem 1. Let (Φ,D) be an information algebra. 1. (Φ,D) is continuous (resp. compact) if, and only if (Φ,≤) is a continuous lattice(resp. algebraic lattice). 2. (Φ,D) is s-continuous (resp. s-compact) if, and only if (Φ,≤) is a complete lattice and for all φ∈Φ, x∈D, φ⇒x =∨{ψ ∈Φ:ψ =ψ⇒x ≪φ}.