Categorical Accommodation of Graded Fuzzy Topological System, Graded Frame and Fuzzy Topological Space with Graded inclusion
aa r X i v : . [ m a t h . G M ] A p r Categorical Accommodation of Graded FuzzyTopological System, Graded Frame and FuzzyTopological Space with Graded inclusion
Purbita Jana a, ∗ , Mihir .K. Chakraborty b a Department of Pure Mathematics, University of Calcutta,35, Ballygunge Circular Road, Ballygunge, Kolkata-700019, West Bengal, India. b School of Cognitive Science, Jadavpur University,188, Raja S. C. Mallick Road, Kolkata-700032, West Bengal, India.
Abstract
A detailed study of graded frame, graded fuzzy topological system and fuzzytopological space with graded inclusion is already done in our earlier paper.The notions of graded fuzzy topological system and fuzzy topological spacewith graded inclusion were obtained via fuzzy geometric logic with graded con-sequence. As an off shoot the notion of graded frame has been developed. Thispaper deals with a detailed categorical study of graded frame, graded fuzzytopological system and fuzzy topological space with graded inclusion and theirinterrelation.
Keywords:
Fuzzy relations, Non-classical logic, Category theory, Topology,Algebra, Graded frame, Graded fuzzy topological system, Fuzzy topologicalspace with graded inclusion, Fuzzy geometric logic with graded consequence.
1. Introduction
The motivation of this work mostly came from the main topic of Vickers’sbook “Topology via Logic” [14], where he introduced a notion of topologicalsystem and indicated its connection with geometric logic. The relationshipsamong topological space, topological system, frame and geometric logic playan important role in the study of topology through logic (geometric logic).Naturally the question “from which logic fuzzy topology can be studied?” comesin mind. If such a logic is obtained what could be its significance?To answer these questions, as basic steps, we first introduced some notionsof fuzzy topological systems and established the interrelation with appropriate ∗ Corresponding author
Email addresses: [email protected] (Purbita Jana), [email protected] (Mihir .K. Chakraborty) The research work of the first author is supported by “Women Scientists Scheme- A(WOS-A)” under the File Number: SR/WOS-A/PM-1010/2014.
Preprint submitted to Fuzzy Sets and Systems November 6, 2018 opological spaces and algebraic structures. These relationships were studied incategorical framework [5, 7, 8, 9, 10, 11, 12].In [4], another level of generalisation (i.e., introduction of many-valuedness)has taken place giving rise to graded fuzzy topological system (vide Definition2.3) and fuzzy topological space with graded inclusion (vide Definition 2.1).It was also required to generalise the notion of frame to graded frame (videDefinition 2.2).Geometric logic is endowed with an informal observational semantics [13]:whether what has been observed does satisfy (match) an assertion or not. Infact, from the stand point of observation, negative and implicational propo-sitions and universal quantification face ontological difficulties. On the otherhand arbitrary disjunction needs to be included (cf. [13] for an elegant dis-cussion on this issue). Now, observations of facts and assertions about themmay corroborate with each other partially. It is a fact of reality and in suchcases it is natural to invoke the concept of ‘satisfiability to some degree of astatement with respect to the observation’. As a result the question whethersome assertion follows from some other assertion might not have a crisp answer‘yes’/‘no’. It is likely that in general the ‘relation of following’ or more techni-cally speaking, the consequence relation turnstile ( ⊢ ) may be itself many-valuedor graded (vide Definition 1.5). For an introduction to the general theory ofgraded consequence relation we refer to [2, 3]. Thus, we have adopted gradedsatisfiability as well as graded consequence in [4].A further generalised notions such as fuzzy geometric logic with gradedconsequence, fuzzy topological spaces with graded inclusion, graded frame andgraded fuzzy topological systems came into the picture [4].The categories of the fuzzy topological spaces with graded inclusion, gradedfuzzy topological systems and graded frames are introduced in this paper. Ontop of that, this paper depicts the transformation of morphisms between theobjects which play an interesting role too. Through the categorical study itbecomes more clear why the graded inclusion is important in the fuzzy topologyto establish the desired connections. Fuzzy geometric logic with graded consequence[4]
The alphabet of the language L of fuzzy geometric logic with graded con-sequence comprises of the connectives ∧ , W , the existential quantifier ∃ , paren-theses ) and ( as well as: • countably many individual constants c , c , . . . ; • denumerably many individual variables x , x , . . . ; • propositional constants ⊤ , ⊥ ; • for each i >
0, countably many i -place predicate symbols p ij ’s, includingat least the 2-place symbol “=” for identity; • for each i >
0, countably many i -place function symbols f ij ’s.2 efinition 1.1 (Term) . Terms are recursively defined in the usual way. • every constant symbol c i is a term; • every variable x i is a term; • if f j is an i -place function symbol, and t , t , . . . , t i are terms then f ij t t . . . t i is a term; • nothing else is a term. Definition 1.2 (Geometric formula) . Geometric formulae are recursivelydefined as follows: • ⊤ , ⊥ are geometric formulae; • if p j is an i -place predicate symbol, and t , t , . . . , t i are terms then p ij t t . . . t i is a geometric formula; • if t i , t j are terms then ( t i = t j ) is a geometric formula; • if φ and ψ are geometric formulae then ( φ ∧ ψ ) is a geometric formula; • if φ i ’s ( i ∈ I ) are geometric formulae then W { φ i } i ∈ I is a geometric formula,when I = { , } then the above formula is written as φ ∨ φ ; • if φ is a geometric formula and x i is a variable then ∃ x i φ is a geometricformula; • nothing else is a geometric formula. Definition 1.3 (Interpretation) . An interpretation I consists of • a set D , called the domain of interpretation; • an element I ( c i ) ∈ D for each constant c i ; • a function I ( f ij ) : D i −→ D for each function symbol f ij ; • a fuzzy relation I ( p ij ) : D i −→ [0 ,
1] for each predicate symbol p ij i.e. it isa fuzzy subset of D i . Definition 1.4 (Graded Satisfiability) . Let s be a sequence over D . Let s =( s , s , . . . ) be a sequence over D where s , s , . . . are all elements of D . Let d be an element of D . Then s ( d/x i ) is the result of replacing i ’th coordinateof s by d i.e., s ( d/x i ) = ( s , s , . . . , s i − , d, s i +1 , . . . ). Let t be a term. Then s assigns an element s ( t ) of D as follows: • if t is the constant symbol c i then s ( c i ) = I ( c i ); • if t is the variable x i then s ( x i ) = s i ;3 if t is the function symbol f ij t t . . . t i then s ( f ij t t . . . t i ) = I ( f ij )( s ( t ) , s ( t ) , . . . , s ( t i )).Now we define grade of satisfiability of φ by s written as gr ( s sat φ ), where φ is a geometric formula, as follows: • gr ( s sat p ij t t . . . t i ) = I ( p ij )( s ( t ) , s ( t ) , . . . , s ( t i )); • gr ( s sat ⊤ ) = 1; • gr ( s sat ⊥ ) = 0; • gr ( s sat t i = t j ) = ( if s ( t i ) = s ( t j )0 otherwise ; • gr ( s sat φ ∧ φ ) = gr ( s sat φ ) ∧ gr ( s sat φ ); • gr ( s sat φ ∨ φ ) = gr ( s sat φ ) ∨ gr ( s sat φ ); • gr ( s sat W { φ i } i ∈ I ) = sup { gr ( s sat φ i ) | i ∈ I } ; • gr ( s sat ∃ x i φ ) = sup { gr ( s ( d/x i ) sat φ ) | d ∈ D } .In [0 , ∧ and ∨ to mean min and max respectively - a conventionthat will be followed throughout. The expression φ ⊢ ψ , where φ , ψ are wffs, iscalled a sequent. We now define satisfiability of a sequent. Definition 1.5. gr ( s sat φ ⊢ ψ )= gr ( s sat φ ) → gr ( s sat ψ ), where → :[0 , × [0 , −→ [0 ,
1] is the G¨ o del arrow defined as follows: a → b = ( if a ≤ bb if a > b, for a, b ∈ [0 , gr ( φ ⊢ ψ ) = inf s { gr ( s sat φ ⊢ ψ ) } , where s ranges over all sequences overthe domain D of interpretation. Theorem 1.6.
Graded sequents satisfy the following properties1. gr ( φ ⊢ φ ) = 1 ,2. gr ( φ ⊢ ψ ) ∧ gr ( ψ ⊢ χ ) ≤ gr ( φ ⊢ χ ) ,3. (i) gr ( φ ⊢ ⊤ ) = 1 , (ii) gr ( φ ∧ ψ ⊢ φ ) = 1 ,(iii) gr ( φ ∧ ψ ⊢ ψ ) = 1 , (iv) gr ( φ ⊢ ψ ) ∧ gr ( φ ⊢ χ ) = gr ( φ ⊢ ψ ∧ χ ) ,4. (i) gr ( φ ⊢ W S ) = 1 if φ ∈ S ,(ii) inf φ ∈ S { gr ( φ ⊢ ψ ) } ≤ gr ( W S ⊢ ψ ) ,5. gr ( φ ∧ W S ⊢ W { φ ∧ ψ | ψ ∈ S } ) = 1 ,6. gr ( ⊤ ⊢ ( x = x )) = 1 ,7. gr ((( x , . . . , x n ) = ( y , . . . , y n )) ∧ φ ⊢ φ [( y , . . . , y n ) / ( x , . . . , x n )]) =1,8. (i) gr ( φ ⊢ ψ [ x/y ]) ≤ gr ( φ ⊢ ∃ yψ ) , (ii) gr ( ∃ yφ ⊢ ψ ) ≤ gr ( φ [ x/y ] ⊢ ψ ) ,9. gr ( φ ∧ ( ∃ y ) ψ ⊢ ( ∃ y )( φ ∧ ψ )) = 1 . . Categories: Graded Fuzzy Top, Graded Fuzzy TopSys, GradedFrm and their interrelationshipsDefinition 2.1. Let X be a set, τ be a collection of fuzzy subsets of X s.t.1. ˜ ∅ , ˜ X ∈ τ , where ˜ ∅ ( x ) = 0 and ˜ X ( x ) = 1, for all x ∈ X ;2. ˜ T i ∈ τ for i ∈ I imply S i ∈ I ˜ T i ∈ τ , where S i ∈ I ˜ T i ( x ) = sup i ∈ I { ˜ T i ( x ) } ;3. ˜ T , ˜ T ∈ τ imply ˜ T ∩ ˜ T ∈ τ , where ( ˜ T ∩ ˜ T )( x ) = ˜ T ( x ) ∧ ˜ T ( x ),and ⊆ be a fuzzy inclusion relation between fuzzy sets, which is defined asfollows. For any two fuzzy subsets ˜ T , ˜ T of X , gr ( ˜ T ⊆ ˜ T ) = inf x ∈ X { ˜ T ( x ) → ˜ T ( x ) } , where → is the G¨ o del arrow.Then ( X, τ, ⊆ ) is called a fuzzy topological space with graded inclu-sion . ( τ, ⊆ ) is called a fuzzy topology with graded inclusion over X .It is to be noted that we preferred to use the traditional notation ˜ A to denotea fuzzy set [1]. We list the properties of the members of fuzzy topology withgraded inclusion, as propositions, that would be used subsequently. By routinecheck all the propositions can be verified. Proposition 2.1. gr ( ˜ T ⊆ ˜ T ) = 1 . Proposition 2.2. gr ( ˜ T ⊆ ˜ T ) = 1 = gr ( ˜ T ⊆ ˜ T ) ⇒ ˜ T = ˜ T . Proposition 2.3. gr ( ˜ T ⊆ ˜ T ) ∧ gr ( ˜ T ⊆ ˜ T ) ≤ gr ( ˜ T ⊆ ˜ T ) . Proposition 2.4. gr ( ˜ T ∩ ˜ T ⊆ ˜ T ) = 1 = gr ( ˜ T ∩ ˜ T ⊆ ˜ T ). Proposition 2.5. gr ( ˜ T ⊆ ˜ X ) = 1. Proposition 2.6. gr ( ˜ T ⊆ ˜ T ) ∧ gr ( ˜ T ⊆ ˜ T ) = gr ( ˜ T ⊆ ˜ T ∩ ˜ T ). Proposition 2.7. gr ( ˜ T i ⊆ S i ˜ T i ) = 1 . Proposition 2.8. inf ˜ T i ∈ S { gr ( ˜ T i ⊆ ˜ T ) } = gr ( S S ⊆ ˜ T ) . Proposition 2.9. gr ( ˜ T ∩ S i ˜ T i ⊆ S i ( ˜ T ∩ ˜ T i )) = 1. Proposition 2.10. ˜ T ( x ) ∧ gr ( ˜ T ⊆ ˜ T ) ≤ ˜ T ( x ), for each x .Note 1: In [6], a notion called localic preordered topological space has beendefined which is a 4-tuple ( X, L, τ, P ) where L is a frame, ( X, L, τ ) is an L -valued fuzzy topological space (localic topological space), P is an L -valued fuzzypreorder (reflexive and transitive) on X and P ( x, y ) ∧ ˜ T ( y ) ≤ ˜ T ( x ), for any x, y ∈ X and ˜ T ∈ τ . In our case, we have taken L as [0 ,
1] and usual fuzzytopological space with the exception that the inclusion relation in τ is definedin terms of a fuzzy implication (G¨ o del). That is, the set X has no orderinghere but the topology τ is endowed with a fuzzy ordering relation namely fuzzyinclusion. 5 efinition 2.2 (Graded Frame) . A graded frame is a 5-tuple ( A, ⊤ , ∧ , W , R ),where A is a non-empty set, ⊤ ∈ A , ∧ is a binary operation, W is an operationon arbitrary subset of A , R is a [0 , A satisfyingthe following conditions:1. R ( a, a ) = 1;2. R ( a, b ) = 1 = R ( b, a ) ⇒ a = b ;3. R ( a, b ) ∧ R ( b, c ) ≤ R ( a, c );4. R ( a ∧ b, a ) = 1 = R ( a ∧ b, b );5. R ( a, ⊤ ) = 1;6. R ( a, b ) ∧ R ( a, c ) = R ( a, b ∧ c );7. R ( a, W S ) = 1 if a ∈ S ;8. inf { R ( a, b ) | a ∈ S } = R ( W S, b );9. R ( a ∧ W S, W { a ∧ b | b ∈ S } ) = 1;for any a, b, c ∈ A and S ⊆ A . We will denote graded frame by ( A, R ).Note 2: A graded frame is a localic preordered set [6] with the value set [0 , Definition 2.3. A Graded fuzzy topological system is a quadruple ( X, | = , A, R ) consisting of a nonempty set X , a graded frame ( A, R ) and a fuzzyrelation | = from X to A such that1. gr ( x | = a ) ∧ R ( a, b ) ≤ gr ( x | = b );2. for any finite subset including null set, S , of A , gr ( x | = V S ) = inf { gr ( x | = a ) | a ∈ S } ;3. for any subset S of A , gr ( x | = W S ) = sup { gr ( x | = a ) | a ∈ S } . Definition 2.4 (Spatial) . A graded fuzzy topological system ( X, | = , A, R ) issaid to be spatial if and only if (for any x ∈ X , gr ( x | = a ) = gr ( x | = b )) imply( a = b ), for any a, b ∈ A .Note 3: The localic preordered topological system defined in [6] has a depar-ture from the above notion in that the fuzzy order relation of localic preorderedtopological system is defined on the set X while in Definition 2.3 it is defined onthe set A . That means in [6], the fuzzy preorder relation is defined on the objectset X while in our case the fuzzy preorder is imposed on the set of properties A , additionally we need more conditions on the preorder of the property set toconnect the fuzzy topological system with fuzzy geometric logic.6 .1. Categories Graded Fuzzy TopDefinition 2.5 ( Graded Fuzzy Top ) . The category
Graded Fuzzy Top isdefined thus. • The objects are fuzzy topological spaces with graded inclusion (
X, τ, ⊆ ),( Y, τ ′ , ⊆ ) etc (c.f. Definition 2.1). • The morphisms are functions satisfying the following continuity property:If f : ( X, τ, ⊆ ) −→ ( Y, τ ′ , ⊆ ) and ˜ T ′ ∈ τ ′ then f − ( ˜ T ′ ) ∈ τ . • The identity on (
X, τ, ⊆ ) is the identity function. That this is an Graded Fuzzy Top morphism can be proved. • If f : ( X, τ, ⊆ ) −→ ( Y, τ ′ , ⊆ ) and g : ( Y, τ ′ , ⊆ ) −→ ( Z, τ ′′ , ⊆ ) are mor-phisms in Graded Fuzzy Top , their composition g ◦ f is the compositionof functions between two sets. It can be verified that g ◦ f is a morphismin Graded Fuzzy Top . Proposition 2.11. gr ( ˜ T ⊆ ˜ T ) ≤ gr ( f ( ˜ T ) ⊆ f ( ˜ T )). Proof.
It is known that gr ( ˜ T ⊆ ˜ T ) = inf x ∈ X { ˜ T ( x ) → ˜ T ( x ) } and gr ( f ( ˜ T ) ⊆ f ( ˜ T )) = inf y ∈ Y { f ( ˜ T )( y ) → f ( ˜ T )( y ) } = inf y ∈ Y { sup x ∈ f − ( y ) { ˜ T ( x ) } → sup x ∈ f − ( y ) { ˜ T ( x ) }} .Now, sup x ∈ f − ( y ) { ˜ T ( x ) } → sup x ∈ f − ( y ) { ˜ T ( x ) } = inf x ∈ f − ( y ) { ˜ T ( x ) → sup x ∈ f − ( y ) { ˜ T ( x ) }} and ˜ T ( x ) ≤ sup x ∈ f − ( y ) ˜ T ( x ) forany x ∈ f − ( y ).Therefore ˜ T ( x ) → ˜ T ( x ) ≤ ˜ T ( x ) → sup x ∈ f − ( y ) { ˜ T ( x ) } for any x ∈ f − ( y ).So, inf x ∈ f − ( y ) { ˜ T ( x ) → ˜ T ( x ) } ≤ inf x ∈ f − ( y ) { ˜ T ( x ) → sup x ∈ f − ( y ) { ˜ T ( x ) }} .Now, inf x ∈ X { ˜ T ( x ) → ˜ T ( x ) } ≤ inf x ∈ f − ( y ) { ˜ T ( x ) → ˜ T ( x ) } as f − ( y ) ⊆ X .So, inf x ∈ X { ˜ T ( x ) → ˜ T ( x ) } ≤ inf x ∈ f − ( y ) { ˜ T ( x ) → sup x ∈ f − ( y ) { ˜ T ( x ) }} forany y ∈ Y and consequently, inf x ∈ X { ˜ T ( x ) → ˜ T ( x ) } ≤ inf y ∈ Y { sup x ∈ f − ( y ) { ˜ T ( x ) } → sup x ∈ f − ( y ) { ˜ T ( x ) }} . Definition 2.6 ( Graded Frm ) . The category
Graded Frm is defined thus. • The objects are graded frames (
A, R ), (
B, R ′ ) etc. (c.f. Definition 2.2). • The morphisms are graded frame homomorphisms defined in the followingway: If f : ( A, R ) −→ ( B, R ′ ) then(i) f ( a ∧ a ) = f ( a ) ∧ f ( a );(ii) f ( W i a i ) = sup i { f ( a i ) } ;(iii) R ( a , a ) ≤ R ′ ( f ( a ) , f ( a )). • The identity on (
A, R ) is the identity morphism. That this is an
Graded Frm morphism can be proved. 7 If f : ( A, R ) −→ ( B, R ′ ) and g : ( B, R ′ ) −→ ( C, R ′′ ) are morphismsin Graded Frm , their composition g ◦ f is the composition of gradedhomomorphisms between two graded frames. It can be verified that g ◦ f is a morphism in Graded Frm (vide Proposition 2.12).
Proposition 2.12. If f : ( A, R ) −→ ( B, R ′ ) and g : ( B, R ′ ) −→ ( C, R ′′ ) aremorphisms in Graded Frm then g ◦ f : ( A, R ) −→ ( C, R ′′ ) is a morphism in Graded Frm . Proof.
To show that g ◦ f : ( A, R ) −→ ( C, R ′′ ) is a graded frame homomorphismwe will proceed in the following way.( i ) g ◦ f ( a ∧ a ) = g ( f ( a ∧ a ))= g ( f ( a ) ∧ f ( a )) , as f is a graded frame homomorphism= g ( f ( a )) ∧ g ( f ( a )) , as g is a graded frame homomorphism= g ◦ f ( a ) ∧ g ◦ f ( a ) . ( ii ) g ◦ f ( _ i a i ) = g ( f ( _ i a i ))= g ( sup i { f ( a i ) } ) , as f is a graded frame homomorphism= sup i { g ( f ( a i )) } , as g is a graded frame homomorphism= sup i { g ◦ f ( a i ) } . ( iii ) R ( a , a ) ≤ R ′ ( f ( a ) , f ( a )) , as f is a graded frame homomorphism ≤ R ′′ ( g ( f ( a )) , g ( f ( a ))) , as g is a graded frame homomorphism= R ′′ ( g ◦ f ( a ) , g ◦ f ( a )) . Definition 2.7 ( Graded Fuzzy TopSys ) . The category
Graded Fuzzy TopSys is defined thus. • The objects are graded fuzzy topological systems ( X, | = , A, R ), ( Y, | = , B, R ′ )etc. (c.f. Definition 2.3). • The morphisms are pair of maps satisfying the following continuity prop-erties: If ( f , f ) : ( X, | = , A, R ) −→ ( Y, | = ′ , B, R ′ ) then(i) f : X −→ Y is a set map;(ii) f : ( B, R ′ ) −→ ( A, R ) is a graded frame homomorphism;(iii) gr ( x | = f ( b )) = gr ( f ( x ) | = ′ b ). • The identity on ( X, | = , A, R ) is the pair ( id X , id A ), where id X is the iden-tity map on X and id A is the identity graded frame homomorphism. Thatthis is an Graded Fuzzy TopSys morphism can be proved. • If ( f , f ) : ( X, | = , A, R ) −→ ( Y, | = ′ , B, R ′ ) and ( g , g ) : ( Y, | = ′ , B, R ′ ) −→ ( Z, | = ′′ , C, R ′′ ) are morphisms in Graded Fuzzy TopSys , their compo-sition ( g , g ) ◦ ( f , f ) = ( g ◦ f , f ◦ g ) is the pair of composition of8unctions between two sets and composition of graded homomorphismsbetween two graded frames. It can be verified that ( g , g ) ◦ ( f , f ) is amorphism in Graded Fuzzy TopSys . In this subsection, we define various functors, which are required to proveour desired results.
Functor
Ext g from Graded Fuzzy TopSys to Graded Fuzzy Top Definition 2.8.
Let ( X, | = , A, R ) be a graded fuzzy topological system, and a ∈ A . For each a , its extent g in ( X, | = , A, R ) is a mapping ext g ( a ) from X to [0 ,
1] given by ext g ( a )( x ) = gr ( x | = a ). Also ext g ( A ) = { ext g ( a ) } a ∈ A and gr ( ext g ( a ) ⊆ ext g ( a )) = inf { ext g ( a )( x ) → ext g ( a )( x ) } x , for any a , a ∈ A . Lemma 2.1. ( ext g ( A ) , ⊆ ) forms a graded fuzzy topology on X .As a consequence ( X, ext g ( A ) , ⊆ ) forms a graded fuzzy topological space. Lemma 2.2.
If ( f , f ) : ( X, | = ′ , A, R ) −→ ( Y, | = ′′ , B, R ′ ) is continuous then f : ( X, ext g ( A ) , ⊆ ) −→ ( Y, ext g ( B ) , ⊆ ) is continuous. Proof. ( f , f ) : ( X, | = ′ , A, R ) −→ ( Y, | = ′′ , B, R ′ ) is continuous.So we have for all x ∈ X, b ∈ B , gr ( x | = ′ f ( b )) = gr ( f ( x ) | = ′′ b ). N ow, ( f − ( ext g ( b )))( x ) = ext g ( b )( f ( x ))= gr ( f ( x ) | = ′′ b )= gr ( x | = ′ f ( b ))= ext g ( f ( b ))( x ) .So, f − ( ext g ( b )) = ext g ( f ( b )) ∈ ext g ( A ). So, f is a continuous map from ( X, ext g ( A ) , ⊆ ) to ( Y, ext g ( B ) , ⊆ ). Definition 2.9. Ext g is a functor from Graded Fuzzy TopSys to Graded Fuzzy Top defined as follows.
Ext g acts on an object ( X, | = ′ , A, R ) as Ext g ( X, | = ′ , A, R ) = ( X, ext g ( A ) , ⊆ )and on a morphism ( f , f ) as Ext g ( f , f ) = f .The above two Lemmas 2.1 and 2.2 show that Ext g is a functor. Functor J g from Graded Fuzzy Top to Graded Fuzzy TopSysDefinition 2.10. J g is a functor from Graded Fuzzy Top to Graded Fuzzy TopSys defined as follows. J g acts on an object ( X, τ, ⊆ ) as J g ( X, τ, ⊆ ) = ( X, ∈ , τ, ⊆ ), where gr ( x ∈ ˜ T ) =˜ T ( x ) for ˜ T in τ , and on a morphism f as J g ( f ) = ( f, f − ). Lemma 2.3. ( X, ∈ , τ, ⊆ ) is a graded fuzzy topological system.9 roof. ( τ, ⊆ ) forms a graded frame can be shown using Propositions 2.1-2.9.To show (1) gr ( x ∈ ˜ T ) ∧ gr ( ˜ T ⊆ ˜ T ) ≤ gr ( x ∈ ˜ T ), (2) gr ( x ∈ ˜ T ∩ ˜ T ) = inf { gr ( x ∈ ˜ T ) , gr ( x ∈ ˜ T ) } and (3) gr ( x ∈ S i ˜ T i ) = sup i { gr ( x ∈ ˜ T i ) } let usproceed in the following way.(1) gr ( x ∈ ˜ T ) ∧ gr ( ˜ T ⊆ ˜ T ) ≤ gr ( x ∈ ˜ T ) follows by Proposition 2.10.(2) gr ( x ∈ ˜ T ∩ ˜ T ) = ( ˜ T ∩ ˜ T )( x )= ˜ T ( x ) ∧ ˜ T ( x )= gr ( x ∈ ˜ T ) ∧ gr ( x ∈ ˜ T ) . (3) gr ( x ∈ [ i ˜ T i ) = ( [ i ˜ T i )( x )= sup i { ˜ T i ( x ) } = sup i { gr ( x ∈ ˜ T i ) } Lemma 2.4. J g ( f ) = ( f, f − ) is continuous provided f is continuous. Proof.
We have f : ( X, τ ) −→ ( Y, τ ) and ( f, f − ) : ( X, ∈ , τ ) −→ ( Y, ∈ , τ ).It is enough to show that ˜ T ∈ τ , gr ( x ∈ f − ( ˜ T )) = gr ( f ( x ) ∈ ˜ T ) as Propo-sition 2.11 holds.Now, gr ( x ∈ f − ( ˜ T )) = ( f − ( ˜ T ))( x ) = ˜ T ( f ( x )) = gr ( f ( x ) ∈ ˜ T ).Hence J g ( f ) = ( f, f − ) is continuous.So J g is a functor from Graded Fuzzy Top to Graded Fuzzy TopSys . Functor f m g from Graded Fuzzy TopSys to Graded Frm op Definition 2.11. fm g is a functor from Graded Fuzzy TopSys to Graded Frm op defined as follows. f m g acts on an object ( X, | = , A, R ) as f m g ( X, | = , A, R ) = ( A, R ) and on amorphism ( f , f ) as f m g ( f , f ) = f .It is easy to see that f m g is a functor. Functor S g from Graded Frm op to Graded Fuzzy TopSysDefinition 2.12.
Let (
A, R ) be a graded frame,
Hom g (( A, R ) , ([0 , , R ∗ )) = { graded f rame hom v : ( A, R ) −→ ([0 , , R ∗ ) } , where R ∗ : [0 , × [0 , −→ [0 ,
1] such that R ∗ ( a, b ) = 1 iff a ≤ b and R ∗ ( a, b ) = b iff a > b , that is, G¨ o delarrow. Lemma 2.5. ( Hom g (( A, R ) , ([0 , , R ∗ )) , | = ∗ , A, R ), where ( A, R ) is a gradedframe and gr ( v | = ∗ a ) = v ( a ), is a graded fuzzy topological system.10 roof. First of all, we need to show that gr ( v | = ∗ a ) ∧ R ( a, b ) ≤ gr ( v | = ∗ b ).That is, it will be enough to show that v ( a ) ∧ R ( a, b ) ≤ v ( b ). Now as v is agraded frame homomorphism so R ( a, b ) ≤ R ∗ ( v ( a ) , v ( b )). Hence v ( a ) ∧ R ( a, b ) ≤ R ( a, b ) ≤ R ∗ ( v ( a ) , v ( b )). Now as v ( a ), v ( b ) ∈ [0 ,
1] so either v ( a ) ≤ v ( b ) or v ( a ) > v ( b ). For v ( a ) ≤ v ( b ), v ( a ) ∧ R ( a, b ) ≤ v ( a ) ≤ v ( b ) and for v ( a ) > v ( b ), R ∗ ( v ( a ) , v ( b )) = 0. So for the second case v ( a ) ∧ R ( a, b ) ≤ R ∗ ( v ( a ) , v ( b )) = v ( b ).Hence gr ( v | = ∗ a ) ∧ R ( a, b ) ≤ gr ( v | = ∗ b ).As v is a graded frame homomorphism, v ( a ∧ b ) = v ( a ) ∧ v ( b ) and hence gr ( v | = ∗ a ∧ b ) = gr ( v | = ∗ a ) ∧ gr ( v | = ∗ b ).Similarly, gr ( v | = ∗ W S ) = sup { gr ( v | = ∗ a ) | a ∈ S } , for S ⊆ A . Lemma 2.6. If f : ( B, R ′ ) −→ ( A, R ) is a graded frame homomorphism then( ◦ f, f ) : ( Hom (( A, R ) , ([0 , , R ∗ )) , | = ∗ , A, R ) −→ ( Hom (( B, R ′ ) , ([0 , , R ∗ )) , | = ∗ , B, R ′ ) is continuous. Definition 2.13. S g is a functor from Graded Frm op to Graded Fuzzy TopSys defined as follows. S g acts on an object ( A, R ) as S g (( A, R )) = (
Hom g (( A, R ) , ([0 , , R ∗ )) , | = ∗ , A, R ) and on a morphism f as S g ( f ) = ( ◦ f, f ).Previous two Lemmas 2.5 and 2.6 show that S g is indeed a functor. Lemma 2.7.
Ext g is the right adjoint to the functor J g . Proof Sketch.
It is possible to prove the theorem by presenting the co-unit ofthe adjunction.Recall that J g ( X, τ, ⊆ ) = ( X, ∈ , τ, ⊆ ) and Ext g ( X, | = , A, R ) = ( X, ext g ( A ) , ⊆ ). So, J g ( Ext g ( X, | = , A, R )) = ( X, ∈ , ext ( A ) , ⊆ ).Let us draw the diagram of co-unit Graded Fuzzy TopSys Graded Fuzzy Top J g ( Ext g ( X, | = , A, R )) ( X, | = , A, R ) J g ( Y, τ ′ , ⊆ ) ξ X J g ( f )( ≡ ( f , f − )) ˆ f ( ≡ ( f , f )) Ext g ( X, | = , A, R )( Y, τ ′ , ⊆ ) f ( ≡ f ) Hence co-unit is defined by ξ X = ( id X , ext ∗ g ) i.e. ( X, ∈ , ext g ( A ) , ⊆ ) ( X, | = , A, R ) ξ X ( id X , ext ∗ g ) where ext ∗ g is a mapping from A to ext g ( A ) such that ext ∗ g ( a ) = ext g ( a ), for all a ∈ A .Now by routine check one can conclude that Ext g is the right adjoint to thefunctor J g . 11iagram of the unit of the above adjunction is as follows: Graded Fuzzy Top Graded Fuzzy TopSys ( X, τ, ⊆ ) Ext ( J ( X, τ, ⊆ )) Ext ( Y, | = ′ , B, R ′ ) η X ( ≡ id X )ˆ f ( ≡ f ) ext ( f )( ≡ f ) J ( X, τ, ⊆ )( Y, | = ′ , B, R ′ ) f ( ≡ ( f , f − )) Observation 3.
If a graded fuzzy topological system ( X, | = , A, R ) is spatial thenthe co-unit ξ X becomes a natural isomorphism. Observation 4.
For any graded fuzzy topological space ( X, τ, ⊆ ) , the unit η X is a natural isomorphism. Observation 3 and Observation 4 gives the following theorem.
Theorem 4.1.
Category of spatial graded fuzzy topological systems is equivalentto the category
Graded Fuzzy Top . Lemma 4.1. f m g is the left adjoint to the functor S g . Proof Sketch.
It is possible to prove the theorem by presenting the unit of theadjunction.Recall that S g (( B, R ′ )) = ( Hom g (( B, R ′ ) , ([0 , , R ∗ )) , | = ∗ , B, R ′ ) where, gr ( v | = ∗ a ) = v ( a ).Hence S g ( f m g ( X, | = , A, R )) = ( Hom g (( A, R ) , ([0 , , R ∗ )) , | = ∗ , A, R ). Graded Fuzzy TopSys Graded Frm op ( X, | = , A, R ) S g ( f m g ( X, | = , A, R )) S g (( B, R ′ )) η A f ( ≡ ( f , f )) S g ˆ f f m g ( X, | = , A, R )( B, R ′ ) ˆ f ( ≡ f ) Then unit is defined by η A = ( p ∗ , id A ). i.e. ( X, | = , A, R ) S g ( f m g ( X, | = , A, R )) η A ( p ∗ , id A ) where p ∗ : X −→ Hom g (( A, R ) , ([0 , , R ∗ )), x p x : ( A, R ) −→ ([0 , , R ∗ ) such that p x ( a ) = gr ( x | = a ) and R ( a, a ′ ) ≤ R ∗ ( p x ( a ) , p x ( a ′ )) for any a, a ′ ∈ A . Now by routine check one can concludethat f m g is the left adjoint to the functor S g . Theorem 4.2.
Ext g ◦ S g is the right adjoint to the functor f m g ◦ J g . roof. Follows from the combination of the adjoint situations in Lemmas 2.7,4.1.The obtained functorial relationships can be illustrated by the followingdiagram:
Graded Fuzzy TopSysGraded Fuzzy Top Graded Frm op fm g ◦ J g Ext g ◦ S g J g Ext g fm g S g
5. Concluding Remarks and Future Directions • This work can be considered as a continuation of the work done in [4]. Theobjects of the categories, considered here, played the role of model of fuzzygeometric logic with graded consequence. These concepts are actually thebyproduct of the study of fuzzy topology via fuzzy geometric logic withgraded consequence [4]. • All the results of this paper may be obtained if we consider the value setas totally ordered frame instead of [0 , • It should be noted that the inclusion relation is defined using G¨ o del arrow.Hence there is a space to change the arrow to define graded inclusion andmake appropriate graded algebraic structures to get the similar connec-tions of this work. Although we do not delve into this matter in this workbut hope to deal with this in near future. • Considering the value set as any frame will give very interesting resultssuch as,1. For graded fuzzy topological space with graded inclusion if we takethe value set any frame L instead of [0 ,
1] and define inclusion relationusing the fuzzy arrow viz. → : L × L −→ L such that a → b = ( L if a ≤ bb otherwise , for a, b ∈ L , where 1 L is the top element of L ,then Proposition 2.6 does not hold. But note that all other proposi-tions shall hold good. The fuzzy arrow defined here will be denotedby G¨ o del like arrow over L .13. In case of graded frame if we consider the relation R , an L -valuedfuzzy relation, together with all the conditions other than R ( a, b ) ∧ R ( a, c ) = R ( a, b ∧ c ) and include the condition R ( a, b ) ∧ R ( a, c ) ≥ R ( a, b ∧ c ), then we may get the categorical connection with theabove described space.3. It is to be noted that if we consider L -fuzzy subset ( L is any frame) of D i , where D is the domain of interpretation, as the interpretation ofeach i -ary predicate symbols and define the validity of a sequent usingG¨ o del like arrow over L then the introduction rule for conjunctionwill be not valid.Note that the mathematical structures involved here will become modelsof fuzzy geometric logic with graded consequence without the introductionrule for conjunction. Fuzzy geometric logic without the introduction rulefor conjunction is itself an interesting issue to study. The work in thisdirection is in our future agenda. Acknowledgements.
The research work of the first author is supported by “WomenScientists Scheme- A (WOS-A)” under the File Number: SR/WOS-A/PM-1010/2014. This research was also supported by the
Indo-European ResearchTraining Network in Logic (IERTNiL) funded by the
Institute of MathematicalSciences, Chennai , the
Institute for Logic, Language and Computation of the
Universiteit van Amsterdam and the
Fakult¨at f¨ur Mathematik, Informatik undNaturwissenschaften of the
Universit¨at Hamburg . References [1] M. K. Chakraborty and T. M. G. Ahsannullah,
Fuzzy topology on fuzzy setsand tolerance topology , Fuzzy Sets and Systems, , 1992, pp. 103–108.[2] M. K. Chakraborty, Graded consequence: further studies , J. Appl. Non-Classical Logics, , 1995, pp. 127–137.[3] M. K. Chakraborty and S. Dutta,
Graded consequence revisited , Fuzzy Setsand Systems, , 2010, pp. 1885–1905.[4] M. K. Chakraborty and P. Jana,
Fuzzy Topology via Fuzzy Geometric Logicwith Graded Consequence , International Journal of Approximate Reason-ing, Elsevier (accepted).[5] J. T. Denniston, A. Melton, and S. E. Rodabough,
Interweaving algebraand topology: Lattice-valued topological systems , Fuzzy Sets and Systems, , 2012, pp. 58–103.[6] J. T. Denniston, A. Melton, and S. E. Rodabough, S. A. Solovyov,
Lattice-valued preordered sets as lattice-valued topological systems , Fuzzy Sets andSystems, , 2015, pp. 89–110. 147] P. Jana and M. K. Chakraborty,
Categorical relationships of fuzzy topolog-ical systems with fuzzy topological spaces and underlying algebras , Ann. ofFuzzy Math. and Inform., , 2014, no. 5, pp. 705–727.[8] P. Jana and M. K. Chakraborty, On Categorical Relationship among vari-ous Fuzzy Topological Systems, Fuzzy Topological Spaces and related Alge-braic Structures , Proceedings of the 13th Asian Logic Conference, WorldScientific, 2015, pp. 124–135.[9] P. Jana and M. K. Chakraborty,
Categorical relationships of fuzzy topolog-ical systems with fuzzy topological spaces and underlying algebras-II , Ann.of Fuzzy Math. and Inform., , 2015, no. 1, pp. 123–137.[10] S. E. Rodabaugh, A categorical accommodation of various notions of fuzzytopology , Fuzzy Sets and Systems, , 1983, pp. 241–265.[11] S. Solovyov, Variable-basis topological systems versus variable-basis topo-logical spaces , Soft Comput., , 2010, no. 10, pp. 1059–1068.[12] S. Solovyov, Localification of variable-basis topological systems , Quaest.Math., , 2011, no. 1, pp. 11–33.[13] S. J. Vickers, Issues of logic, algebra and topology in ontology , in: R. Poli,M. Healy, A. Kameas (Eds.), Theory and Applications of Ontology: Com-puter Applications, volume 2 of Theory and Applications of Ontology, 2010.[14] S. J. Vickers,