Formal Concepts and Residuation on Multilattices}
aa r X i v : . [ m a t h . L O ] J un FORMAL CONCEPTS AND RESIDUATION ON MULTILATTICES *BLAISE B. KOGUEP NJIONOU, **LEONARD KWUIDA, ***CELESTIN LELEA
BSTRACT . Let A i : “ p A i , ď i , J i , d i , Ñ i , K i q , i “ , be two complete residu-ated multilattices, G (set of objects) and M (set of attributes) be two nonemptysets and p ϕ, ψ q a Galois connection between A G and A M . In this work weprove that C : “ tp h, f q P A G ˆ A M | ϕ p h q “ f and ψ p f q “ h u is a complete residuated multilattice. This is a generalization of a result byRuiz-Calviño and Medina [RM12] saying that if the (reduct of the) algebras A i , i “ , are complete multilattices, then C is a complete multilattice.
1. I
NTRODUCTION : FCA
AND F UZZYFICATION
In the theory of fuzzy concept analysis, the underlying set of truth valuesis generally a lattice. Medina and Ruiz-Calviño proposed in [RM12] a newapproach of fuzzy concept analysis by using multilattices as underlying setof truth values. Multilattices were introduced by Benado [Bm55]. The mainidea is to relax the requirement on the existence of least upper bounds andgreatest lower bounds, and to ask only that the set of minimal upper boundsand the set of maximal lower bounds of any pair of elements are non-empty.In this paper, we show that residuation on multilattices can be useful in fuzzyconcept analysis to evaluate the attributes and objects. Especially, we willuse the adjoint pair in a residuated multilattice to built a residuated conceptmultilattice. We organize the present contribution as follows: in Section 2 werecall some notions of Formal Concept Analysis and multilattices. In Section 3,we apply the ordinary sum construction to create a new residuated multilatticefrom the old ones and prove that every bounded pure residuated multilatticehas order equal or greater than . We also briefly show how Medina and Ruiz-Calviño used multilattices as truth degree sets. Section 4 is .2. C ONCEPT L ATTICES AND M ULTILATTICES A formal context is set up with the sets G of objects, M of attributes and abinary relation I Ď G ˆ M . We denote it by K : “ p G, M, I q , and write p g, m q P Ior g I m to mean that the object g has the attribute m . In the whole paperwe assume that G and M are non-empty. To extract knowledge from formalcontexts, one can get clusters of objects/attributes called concepts. The Port-Royal Logic School considers a concept as defined by two parts: an extentand an intent. The extent contains all entities belonging to the concept andthe intent is the set of all attributes common to all entities in the concept. Toformalize the notion of concept we need the derivation operator , defined on A Ď G and B Ď M by: A : “ t m P M | g I m for all g P A u and B : “ t g P G | g I m for all m P B u . A formal concept is a pair p A, B q with A “ B and B “ A . The set of formalconcepts of K is denoted by B p K q . It forms a complete lattice, called conceptlattice of the context K , when ordered by the concept hierarchy below: p A , B q ď p A , B q : ðñ A Ď A . Recall that a lattice is a poset in which all finite subsets have a least upperbound and a greatest lower bound. A lattice is complete if every subset has aleast upper bound and a greatest lower bound. For any subset X of L we denoteby Ž X its least upper bound and by Ź X its greatest lower bound, wheneverthey exist. For X “ t x, y u we write x _ y : “ Ž X and x ^ y : “ Ź X .For any context p G, M, I q the pair p , q forms a Galois connection between P p G q and P p M q and c : X ÞÑ X a closure operator (on P p G q or P p M q , orderedby the inclusion). Recall that a closure operator on a poset p P, ďq is a map c : P Ñ P such that x ď c p y q ðñ c p x q ď c p y q for all x, y P P. A pair p ϕ, ψ q is a Galois connection between the two posets p P, ďq and p Q, ďq if ϕ : P Ñ Q and ψ : Q Ñ P are maps such that p ď ψq ðñ q ď ϕp for all p P P and for all q P Q. In general we will call any pair p p, q q a concept if ϕp “ q and ψq “ p , whenever p ϕ, ψ q is a Galois connection. The maps ϕ ˝ ψ and ψ ˝ ϕ (denoted ϕψ and ψϕ for short) are the corresponding closure operators on P and Q respectively. Anelement p P P is closed if ψϕ p p q “ p . Similarly q P Q is closed if ϕψ p q q “ q .Objects, attributes or incidences can be of fuzzy nature. Several sets of truthdegrees have been considered, starting from the interval r , s [ZL65] to lat-tices [GJ67], multilattices [MOR13] and residuations on these structures. In[RM12] Ruiz-Calviño and Medina investigate the use of multilattices as un-derlying set of truth-values for attributes and objects, and prove that the setof all concepts is a multilattice. We will use residuated multilattices as the setof truth-values to evaluate attributes and objects, and show that the set of allconcepts in this case is a residuated multilattice.Multilattices generalize lattices and were introduced by Mihail Benado [Bm55].Let p P, ďq be a poset and x, y P P . We say that x is below y or y is above x ,whenever x ď y . Definition 1. [RM12]
A poset p P, ďq is called coherent if every chain hassupremum and infimum. Definition 2. [MOR07]
A poset p P, ďq is called a multilattice or m-lattice iffor any finite subset X of P , each upper bound of X is above a minimal upperbound of X and each lower bound of X is below a maximal lower bound of X . For any subset X of P , we denote by \ X (resp. [ X ) the set of its minimalupper bounds (resp. maximal lower bounds). In general, \ X and [ X couldbe empty (when they are not empty, the multilattice is said to be full ) . But if p P, ďq is a bounded finite poset, then \ X and [ X are non-empty. For X “ t x, y u We also accept that any element is below and above itself.
CA ON RESIDUATED MULTILATTICES 3 we write x \ y and x [ y instead of \ X and [ X . When \ X or [ X is a singleton itwill be consider as an element of P , i.e., we write \ X P P or [ X P P instead of \ X Ď P or [ X Ď P . If p P, ďq is a lattice and X a finite subset of P , then the setof upper (resp. lower) bounds of X is non-empty and has exactly one minimal(resp. maximal) element, namely _ X (resp. ^ X ). Moreover, any upper (resp.lower) bound of X is above (resp. below) _ X (resp. ^ X ). Thus any lattice is amultilattice with \ X “ t_ X u and [ X “ t^ X u . We call a multilattice pure ifit is not a lattice.FCA can be seen as applied theory of complete lattices. Below is how thecompleteness can be carried out to multilattices. Definition 3. [MOR07]
A poset p P, ďq is a complete multilattice if for all X Ď P , the sets \ X, [ X are non empty and each upper bound of X is above anelement of \ X and each lower bound is below an element of [ X . K a bdc J K a bdcF
IGURE
1. A complete multilattice (left) and a non completemultilattice (right).On Figure 1 we can see the smallest bounded poset that is a complete multi-lattice but not a lattice. We will usually refer to it as the multilattice ML . Allfinite lattices are complete lattices, and all finite bounded posets are completemultilattices.In the framework of multilattices, the concept of homomorphism has beenoriginally introduced by M. Benado [Bm55]. Definition 4. [Bm55]
A map h : P Ñ P between multilattices is said to be a homomorphism if h p x \ y q Ď h p x q \ h p y q and h p x [ y q Ď h p x q [ h p y q , for all x, y P P . When the initial multilattice is full, the notion of homomorphism can becharacterized in terms of equalities.
Proposition 1. [CCMO14]
Let h : P Ñ P be a map between multilatticeswhere P is full. Then h is a homomorphism if and only if, for all x , y P P , h p x \ y q “ p h p x q \ h p y qq X h p P q and h p x [ y q “ p h p x q [ h p y qq X h p P q . A homomorphism h will be called an isomorphism when it is one-to-onehomomorphism. In [RM12] the authors defined complete multilattices as coherent posets p P, ďq with no infiniteantichain and \ X ‰ H ‰ [ X for all X Ď P . KOGUEP, KWUIDA, LELE
In [MOR07], J.Medina, M. Ojeda-Aciego and J. Ruiz-Calviño defined two dif-ferent types of submultilattices, namely : full submultilattice (or f-submultilattice)and restricted submultilattice (or r-submultilattice).
Definition 5. [MOR07]
Let p P, ďq be a multilattice and X be a nonempty subsetof P . (i) X is called a full submultilattice ( f - submultilattice ) of P if for all x, y P X , x \ y Ď X and x [ y Ď X (SML-1). (ii) X is called a restricted submultilattice ( r -submultilattice ) of P iffor all x, y P X , p x \ y q X X ‰ H and p x [ y q X X ‰ H (SML-2). It was proved in [MOR07] that if X is a full submultilattice of a multilat-tice P , then equipped with the restriction of the partial order from P , X isa multilattice on its own right. However, this is not the case for restrictedsubmultilattices. Proposition 2.
Every pure and bounded multilattice contains a restricted sub-multilattice isomorphic to ML .Proof. Let p P, ďq be a pure and bounded multilattice. Then, there exists atleast two elements x, y P P such that x \ y is not a singleton. Let a, b P x \ y with a ‰ b . If a \ b and x [ y are singleton, it is enough to choose t x [ y, x, y, a, b, a \ b u as an isomorphic copy of ML , which is indeed a full submultilattice of P .Otherwise, fix an element z P x [ y and another element c P a \ b . Then t z, x, y, a, b, c u is a restricted submultilattice which is isomorphic to ML . (cid:3) A pure and bounded multilattice does not always contain a full submultilat-tice isomorphic to ML , as the following example shows.Consider the multilattice depicted in the following Hasse diagram. J h sssssss i j ❑❑❑❑❑❑❑ d ttttttt f ttttttt❏❏❏❏❏❏❏ g ❏❏❏❏❏❏❏ a sssssss b sssssss❑❑❑❑❑❑❑ c ❑❑❑❑❑❑❑ K ▲▲▲▲▲▲▲ rrrrrrr F IGURE
2. A pure bounded multilattice with no full submultilattice .All the restricted submultilattices of this multilattice isomorphic to ML are: S “ tK , a, b, d, f, i u ; S “ tK , a, b, d, f, h u ; S “ tK , b, c, f, g, i u ; S “ tK , b, c, f, g, j u ; S “ t a, h, i, d, f, Ju ; S “ t b, d, f, h, i, Ju ; S “ t b, f, g, i, j, Ju ; S “ t c, f, g, i, j, Ju ; S “ tK , a, c, j, h, Ju . None of these is a full submultilattice.Now that we know what a complete multilattice is, we are ready to introduceresiduated multilattices. These will serve as truth degree sets to evaluateobjects and attributes. CA ON RESIDUATED MULTILATTICES 5
3. R
ESIDUATION AND T RUTH D EGREE S ETS
The structures of truth values in fuzzy logic and fuzzy set theory are usuallylattices or residuated lattices (see [GJ67, HP98, HU96]). Particularly in fuzzyconcept analysis, residuated lattices are used to evaluate the attributes andobjets. But the necessity of the use of a more general structure arise in someexamples. In [RM12] multilattices are used as the underlying set of truth-values in FCA.
Definition 6. [JT02]
A structure A : “ p A, ď , J , d , Ñq is called pocrim (par-tially ordered commutative residuated integral monoid) if p A, ď , Jq is a posetwith a maximum J and p A, d , Jq is a commutative monoid such that for all a, b, c P A , p;q a d c ď b ðñ c ď a Ñ b. Any pair pd , Ñq satisfying p;q is called a residuated couple or an adjointcouple on p A, ďq . The following properties hold in any pocrim and for the properties of com-plete residuated lattices, we refer to [JT02] A .For all a, b, c P A , P1 a d b ď a and a d b ď b ; P2 " a d p a Ñ b q ď a ď b Ñ p a d b q b d p a Ñ b q ď b ď a Ñ p a d b q ; P3 a ď b ô a Ñ b “ J ; P4 a ď b ùñ $&% a d c ď b d c,c Ñ a ď c Ñ b,b Ñ c ď a Ñ c ; P5 a Ñ p b Ñ c q “ b Ñ p a Ñ c q ; P6 p a Ñ b q d p b Ñ c q ď a Ñ c ; P7 $&% a Ñ b ď p a d c q Ñ p b d c q a Ñ b ď p c Ñ a q Ñ p c Ñ b q a Ñ b ď p b Ñ c q Ñ p a Ñ c q A pocrim is bounded if it also has a lower bound, usually denoted by K .Residuated multilattices were introduced in [CCMO14]. Definition 7. [CCMO14]
A (bounded) residuated multilattice (write
RML for short) A is a (bounded) pocrim whose underlying poset p A, ďq is a multilat-tice. A complete residuated multilattice is a pocrim whose underlying posetis a complete multilattice. To obtain a general procedure to construct new residuated multilattices, wewill extend the ordinal sum construction of pocrims to residuated multilattices.First, we recall the construction of the ordinal sum of pocrims. Let p A, ď A , d A , Ñ A , J A q and p B, ď B , d B , Ñ B , J B q be two pocrims. Consider the set C : “p A > B q{tJ A ” J B u (that is the disjoint union of A and B with J A and J B identified) and define ď on C by x ď y if x, y P A and x ď A y , or x, y P B and KOGUEP, KWUIDA, LELE x ď B y or x P A ztJ A u and y P B . Define d and Ñ on C by: x d y “ $’’’&’’’% x d A y if x, y P Ax d B y if x, y P Bx if x P A ztJ A u and y P By if y P A ztJ A u and x P Bx Ñ y “ $’’’&’’’% x Ñ A y if x, y P Ax Ñ B y if x, y P B J B if x P A ztJ A u and y P By if y P A ztJ A u and x P B Then p C, ď , d , Ñ , K A , J B q is a pocrim, called the ordinal sum of A and B anddenoted by A ‘ B . Note that in A ‘ B , A ztJ A u is below every element of B . Ourgoal is to apply this construction to create new RML from old ones. The firststep in this process consists in identifying the smallest pure
RML .In what follows in this part, our aim is to prove by contradiction that thereis no pocrim structure on ML and to give an example of pocrim structure on apure complete multilattice with elements. Lemma 1.
Suppose that the multilattice ML is endowed with a pocrim struc-ture p ML , ď , d , Ñ , K , Jq . Then the implication Ñ is given by Table below.Proof. We shall use repeatedly the fact that in any pocrim y ď x Ñ y and x Ñ p y Ñ z q “ y Ñ p x Ñ z q hold.(i) Since c ď d Ñ c ‰ J , then d Ñ c “ c . Similarly c Ñ d “ d .(ii) Note that since b ď c Ñ b and b ď d Ñ b , then c Ñ b P t b, c, d u and d Ñ b P t b, c, d u . We narrow down the possibilities further. Note that c Ñ b ď c Ñ d “ d , hence c Ñ b ‰ c . Likewise, d Ñ b ‰ d . Thus c Ñ b P t b, d u and d Ñ b P t b, c u . We show that c Ñ b “ b and d Ñ b “ b by showing that the other combinations lead to contradiction. – If c Ñ b “ b and d Ñ b “ c , then c “ d Ñ b “ d Ñ p c Ñ b q “ c Ñ p d Ñ b q “ c Ñ c “ J , which is acontradiction. – If c Ñ b “ d and d Ñ b “ b , then J “ d Ñ d “ d Ñ p c Ñ b q “ c Ñ p d Ñ b q “ c Ñ b , which impliesthat c ď b . This is impossible. – If c Ñ b “ d and d Ñ b “ c , then J “ a Ñ c “ a Ñ p d Ñ b q “ d Ñ p a Ñ b q . So d ď a Ñ b ‰ J , hence d “ a Ñ b . Now, J “ a Ñ d “ a Ñ p c Ñ b q “ c Ñ p a Ñ b q “ c Ñ d ,which is again a contradiction.Since a and b play symmetrical roles, we deduce c Ñ a “ a and d Ñ a “ a .(iii) Since b ď a Ñ b , then a Ñ b P t b, c, d u . Again, we show that a Ñ b P t c, d u is impossible. Indeed, if a Ñ b “ c , then J “ c Ñ c “ c Ñ p a Ñ b q “ a Ñ p c Ñ b q “ a Ñ b , which is impossible.Similarly, if a Ñ b “ d , then J “ d Ñ d “ d Ñ p a Ñ b q “ a Ñ p d Ñ b q “ CA ON RESIDUATED MULTILATTICES 7 a Ñ b , which is the same contradiction. Hence, a Ñ b “ b . The proofthat b Ñ a “ a is analogous.(iv) It remains to show that a Ñ K “ b, b
Ñ K “ a and c Ñ K “ d Ñ K “ K .Note that a d b ď a, b , so a d b “ K . Thus, a ď b Ñ K and b ď a Ñ K .On the other hand, b Ñ K ď b Ñ a “ a . Hence, b Ñ K “ a and asimilar argument shows that a Ñ K “ b . Finally, observe that a, b ď c ,so c Ñ K ď a Ñ K , b
Ñ K . Hence, c Ñ K ď b, a and consequently c Ñ K “ K . The verification that d Ñ K “ K is similar.We have justified all the entries of the following table. (cid:3)
Ñ K a b c d
JK J J J J J J a b J b J J J b a a
J J J J c K a b J d J d K a b c J JJ K a b c d J T ABLE Proposition 3.
There does not exist a residuated multilattice structure on ML extending its existing partial order.Proof. By contradiction suppose that there exist d and Ñ such that p ML , ď , d , Ñ , K , Jq is a residuated multilattice. Then by Lemma 1, Ñ is given bytable 1 above.Since a d a ď a , we have a d a P tK , a u .If a d a “ K , then a d a ď b and a ď a Ñ b “ b (Table 1), which is acontradiction.Suppose a d a “ a . Since a ď c , we have a “ a d a ď a d c ď a . Hence, a d c “ a . From a ď d , we have a “ c d a ď c d d ď c, d . Because c and d areincomparable, c d d “ a . It follows that d ď c Ñ a “ a (Table 1), which is againa contradiction. (cid:3) Combining Proposition 2 and Proposition 3 we obtain that every boundedpure
RML has order greater than or equal to . The next example shows thatthere is indeed a bounded pure RML of order , which shall subsequently bedenoted by RML , with the operations d and Ñ defined in the following tables.Now that we have setup a smaller pure RML , we can give the structure onthe set of all mappings from a non-empty set to a residuated multilattice.Given a residuated multilattice A and a nonempty set X , we will denoteby A X the set of all mappings from X to A . An order on A X as well as theoperations d , Ñ are defined pointwise: i.e., for f , f P A X we have ‚ f ď f : ðñ f p x q ď f p x q for all x P X , ‚ p f d f q p x q : “ f p x q d f p x q , for any x P X , ‚ p f Ñ f q p x q : “ f p x q Ñ f p x q , for any x P X . Lemma 2.
Let A be a complete residuated multilattice and X a nonempty set.Then ` A X , ď , d , Ñ , J , K ˘ is a complete residuated multilattice. KOGUEP, KWUIDA, LELE K a bdc e J F IGURE
3. A complete residuated multilattice. d K a b c d e
JK K K K K K K K a K a K K a K ab K K b b
K K bc K K b c
K K cd K a K K d K de K K K K K a e
J K a b c d e
J Ñ K a b c d e
JK J J J J J J J a b J b J J J J b a a
J J J J J c K a b J d J J d K a b c J J J e d e d e d
J JJ K a b c d e J T ABLE
2. Of d (left) and Ñ (right) on RML Proof.
Trivially, ď is an order relation on A X . The map Ú : x ÞÑ J is the greatestelement of A X and the map Û : x ÞÑ K is the smallest element of A X . For f , f , f P A X , we have f d f ď f ðñ p f d f qp x q ď f p x q , @ x P X ðñ f p x q d f p x q ď f p x q , @ x P X ðñ f p x q ď f p x q Ñ f p x q , @ x P X ðñ f ď f Ñ f We still need to define \ and [ on A X . For all x P X , the sets f p x q \ f p x q and f p x q [ f p x q are non-empty. By the axiom of choice, define mappings f \ : x ÞÑ f \ p x q P f p x q \ f p x q and f [ : x ÞÑ f [ p x q P f p x q [ f p x q . Each map f \ (resp. f [ ) is a minimal upper (resp. maximal lower) bound of t f , f u .Generally, given t f i | i P I u a non-empty subset of A X , for all x P X , \t f i p x q| i P I u and [t f i p x q| i P I u exist and they are non-empty. Hence, bythe axiom of choice, we can define the mappings f \ : x ÞÑ f \ p x q P \t f i p x q| i P I u and f [ : x ÞÑ f [ p x q P [t f i p x q| i P I u . (cid:3) Let p A , ď q and p A , ď q be two complete multilattices, G and M two non-empty sets (of objects and attributes), and p ϕ, ψ q a Galois connection between p A G , ď q and p A M , ď q . Recall that a concept is a pair p h, f q P A G ˆ A M suchthat ψ p f q “ h and ϕ p h q “ f . Concepts are ordered by p h , f q ď p h , f q : ðñ h ď h p or equivalently f ě f q . CA ON RESIDUATED MULTILATTICES 9
Theorem 1.
Let p A i , ď i q , i “ , be two complete multilattices, G and M betwo non-empty sets and p ϕ, ψ q be a Galois connection between p A G , ď q and p A M , ď q . Let H Ď A G , F Ď A M and C the set of concepts of A G ˆ A M . Then (i) [ ψ p F q Ď ψ p\ F q and [ ϕ p H q Ď ϕ p\ H q . (ii) The poset p C , ďq of all concepts of A G ˆ A M is a complete multilattice,with: Ű j P J p h j , f j q : “ tp h, ϕ p h qq ; h P [t h j | j P J uu and Ů j P J p h j , f j q : “ tp ψ p f q , f q ; f P [t f j ; | j P J uu .Proof. (i) Let h P [ ψ p F q ; i.e. h is a maximal lower bound of ψ p F q . For all f P F , we have h ď ψ p f q , and ϕh ě ϕψ p f q ě f . Thus ϕh is an upperbound of F , and is by then above a minimal upper bound of F . Thusthere is f P \ F such that ϕh ě f . For any f P F we have f ě f and h ď ψϕ p h q ď ψ p f q ď ψ p f q . Thus h, ψϕh and ψf are all lowerbounds of ψ p F q . From the maximality of h we get h “ ψϕh “ ψf , and h P ψ p\ F q . Thus [ ψ p F q Ď ψ p\ F q . The inclusion [ ϕ p H q Ď ϕ p\ H q follows immediately.(ii) Let C J : “ tp h j , f j q | j P J u be a set of concepts. We need to showthat C J has a maximal lower bound, and that any lower bound of C J isbelow a maximal lower bound of C J . First, because [t h j | j P J u and [t f j ; | j P J u are non-empty, the sets tp h, ϕ p h qq ; h P [t h j | j P J uu and tp ψ p f q , f q ; f P [t f j ; | j P J uu are non-empty. Let h P [t h j | j P J u . Wehave h ď h j and f j “ ϕ p h j q ď ϕ p h q , for all j P J . Therefore, p h, ϕ p h qq ďp h j , f j q , for all j P J . That is p h, ϕ p h qq is a lower bound of C J and bythe maximality of h , it is a maximal lower bound of C J . And because h ď ψϕ p h q ď ψϕ p h j q , for all j P J , by the maximality of h , we have ψϕ p h q “ h . Hence, p h, ϕ p h qq is a concept.Similarly, we prove that, for any f P [t f j ; | j P J u , p ψ p f q , f q is aconcept and it is a minimal upper bound of C J . (cid:3) We are going to show that starting with two complete residuated multilat-tices A and A , the set of concepts again forms a complete residuated multi-lattice. 4. R ESIDUATED CONCEPT MULTILATTICES
From now on, A i : “ p A i , ď i , J i , d i , Ñ i , K i q , i “ , are complete residuatedmultilattices, G is a set of objects (to be evaluated in A ), M is a set of at-tributes (to be evaluated in A ), and p ϕ, ψ q is a Galois connection between A G and A M . Further we denote by C the set of concepts, Ext p C q the set of extentsand Int p C q the set of intents. i.e C : “ tp h, f q P A G ˆ A M | ϕ p h q “ f and ψ p f q “ h u , Ext p C q : “ t h P A G | p h, ϕ p h qq P C u “ t h P A G | ψϕ p h q “ h u , Int p C q : “ t f P A M | p ψ p f q , f q P C u “ t f P A M | ϕψ p f q “ f u . The operations d i and Ñ i are defined componentwise on A G and A M . Let Û i and Ú i be the constant maps with values K i and J i : i.e. for g P G and m P M , Û p g q “ K , Û p m q “ K , Ú p g q “ J and Ú p m q “ J . For any h P A G and f P A M we have h d Û “ Û , h d Ú “ h, h Ñ Ú “ Ú and f d Û “ Û , f d Ú “ f, f Ñ Ú “ Ú . We are interested in constructing a residuated couple on p C , ďq ; We are lookingfor a suitable product ( d ) and implication ( Ñ ) such that p h , f q d p h , f q ď p h , f q ðñ p h , f q ď p h , f q Ñ p h , f q . Lemma 3.
Let h , h P A G , f , f P A M , and p ϕ, ψ q a Galois connection. Then (1) ψϕ p h d h q “ min t ψ p f q | f P A M and h d h ď ψ p f qu (2) ϕψ p f d f q “ min t ϕ p h q | h P A G and f d f ď ϕ p h qu (3) ψϕ p h Ñ h q “ max t ψϕ p h q | h P A G and h d h ď h u (4) ϕψ p f Ñ f q “ max t ϕψ p f q | f P A M and f d f ď f u .Proof. Let p ϕ, ψ q be a Galois connection.(1) Let h , h P A G . We set H : “ t ψ p f q | f P A M and h d h ď ψ p f qu . and F “ t f P A M | h d h ď ψ p f qu and get H “ ψ p F q .Since h d h ď ψϕ p Ú q “ Ú , it follows that Ú P H and ϕ p Ú q P F . We claim that H has a least element. In fact, for all ψ p f q P H , h d h ď ψ p f q . Therefore there is a maximal lower bound h of H such that h d h ď h . Thus h P [ H “ [ ψ p F q Ď ψ p\ F q , and there is f P \ F such that h “ ψ p f q . Hence, h P H and h P [ H . Henceforth h is the smallest element of H . Moreover, from h d h ď ψϕ p h d h q we get ϕ p h d h q P F and ψϕ p h d h q P H . It follows that h d h ď ψϕ p h d h q ď ψϕψ p f q “ ψ p f q “ h, and ψϕ p h d h q “ h “ min H .(2) Similarly, the set t ϕ p h q | h P A G and f d f ď ϕ p h qu is non-empty andhas a least element, namely ϕψ p f d f q , for any f , f P A M .(3) Let h , h P A G . We set H “ t ψϕ p h q | h P A G and h d h ď h u . Then H “ t ψϕ p h q | h P A G and h d h ď h u“ t ψϕ p h q | h P A G and h ď h Ñ h uĎ t ψϕ p h q | h P A G and ψϕ p h q ď ψϕ p h Ñ h qu , ψϕ isotone operatorThus ψϕ p h Ñ h q is an upper bound of H . Moreover, h Ñ h ď h Ñ h , then ψϕ p h Ñ h q is in H .Thus, ψϕ p h Ñ h q “ max t ψϕ p h q | h P A G and h ď h Ñ h u .(4) The proof is similar to 3. (cid:3) CA ON RESIDUATED MULTILATTICES 11
Let h , h P A G and f , f P A M . We define the operations b , b , Ñ and Ñ as follows: h b h : “ ψϕ p h d h q h Ñ h : “ ψϕ p h Ñ h q f b f : “ ϕψ p f d f q f Ñ f : “ ϕψ p f Ñ f q . Observe that the operations b and Ñ are defined on all pairs h , h P A G buttheir results are in Ext p C q . Similarly, the operations b and Ñ are definedon all pairs f , f P A M but their results are in Int p C q . We can then infer that p h b h , ϕ p h b h qq “ p ψϕ p h d h q , ϕ p h d h qq is a concept and also that p h Ñ h , ϕ p h Ñ h qq “ p ψϕ p h Ñ h q , ϕ p h Ñ h qq is a concept. Similarly, p ψ p f b f q , f b f q and p ψ p f Ñ f q , f Ñ f q are concepts.It is obvious that the operations b i , i P t , u are commutative. Now, let uslook about their associativity and the residuated couple needed. Lemma 4.
Let p ϕ, ψ q be a Galois connection. If Ext p C q and Int p C q are closedunder Ñ i then pb i , Ñ i q , i “ , are residuated couples and b and b areassociative.Proof. Let h , h , h P A G such that h b h ď h . h b h ď h ùñ h ď ψϕ p h q P t ψϕ p h q | h P A G and h b h ď h uùñ h ď max t ψϕ p h q | h P A G and h b h ď h uùñ h ď max t ψϕ p h q | h P A G and ψϕ p h d h q ď h uùñ h ď max t ψϕ p h q | h P A G and h d h ď h uùñ h ď max t ψϕ p h q | h P A G and h ď h Ñ h uùñ h ď ϕψ p h Ñ h q “ h Ñ h . Henceforth, for all h , h , h in A G we get h b h ď h ùñ h ď h Ñ h . The converse holds if h and h Ñ h are closed. In fact, h ď h Ñ h ðñ h ď ψϕ p h Ñ h qùñ h ď h Ñ h , assuming h Ñ h closed ðñ h d h ď h ùñ ψϕ p h d h q ď h , assuming h closed ðñ h b h ď h . Thus pb , Ñ q is an adjoint couple on Ext p C q , if Ext p C q is closed under Ñ .Similarly, we can proved that pb , Ñ q is an adjoint couple on Int p C q , if itis closed under Ñ . We still need to check the associativity of b and b . Let h , h , h in A G . Then p h b h q b h “ ψϕ pp h b h q d h q“ min t ψ p f q | f P A M and p h b h q d h ď ψ p f qu , and h b p h b h q “ ψϕ p h d p h b h qq“ min t ψ p f q | f P A M and h d p h b h q ď ψ p f qu . It is enough to prove that the sets H “ t ψ p f q | f P A M and p h b h q d h ď ψ p f qu and H “ t ψ p f q | f P A M and h d p h b h q ď ψ p f qu are equal. Let f P A M such that p h b h q d h ď ψ p f q . p h b h q d h ď ψ p f q ùñ p h b h q ď h Ñ ψ p f qùñ ψϕ p h d h q ď h Ñ ψ p f qùñ h d h ď h Ñ ψ p f qùñ p h d h q d h ď ψ p f qùñ ψϕ pp h d h q d h q ď ψ p f qùñ ψϕ p h d p h d h qq ď ψ p f qùñ h b p h d h qq ď ψ p f qùñ h d h ď h Ñ ψ p f qùñ ψϕ p h d h q ď h Ñ ψ p f qùñ h b h ď h Ñ ψ p f qùñ h b p h b h q ď ψ p f qùñ h d p h b h q ď ψ p f q . Thus ψ p f q P H ùñ ψ p f q P H .Conversely, let f P A M such that h d p h b h q ď ψ p f q . h d p h b h q ď ψ p f q ùñ h b h ď h Ñ ψ p f qùñ ψϕ p h d h q ď h Ñ ψ p f qùñ h d h ď h Ñ ψ p f qùñ h d p h d h q ď ψ p f qùñ ψϕ p h d p h d h qq ď ψ p f qùñ ψϕ pp h d h q d h q ď ψ p f qùñ p h d h q b h ď ψ p f qùñ h d h ď h Ñ ψ p f qùñ ψϕ p h d h q ď h Ñ ψ p f qùñ h b h ď h Ñ ψ p f qùñ p h b h q b h ď ψ p f qùñ p h b h q d h ď ψ p f q . Thus ψ p f q P H ùñ ψ p f q P H . Therefore H “ H , and h b p h b h q “ p h b h q b h . (cid:3) CA ON RESIDUATED MULTILATTICES 13
We can now define the product d and the implication Ñ on the set of allconcepts. Theorem 2.
Let A and A be two complete residuated multilattices, G and M two non-empty sets (of objects and attributes) and p ϕ, ψ q a Galois connectionbetween A G and A M . Then C “ tp h, f q P A G ˆ A M | ϕ p h q “ f and ψ p f q “ g uq is a complete residuated multilattice, if Ext p C q and Int p C q are closed under Ñ and Ñ respectively, with p h , f q d p h , f q “ p h b h , ϕ p h b h qq and p h , f q Ñ p h , f q “ p h Ñ h , ϕ p h Ñ h qq . for all concepts p h i , f i q , i “ , .Proof. In [MOR13] it is shown that p C , ĺ q is a complete multilattice. Let p h , f q , p h , f q and p h , f q in C . From Lemma 4 we know that p h b h , ϕ p h b h qq and p h Ñ h , ϕ p h Ñ h qq are concepts. It remains to prove that d and Ñ satisfy the adjointness condition that is, p h , f q d p h , f q ď p h , f q ðñ p h , f q ď p h , f q Ñ p h , f q . This is equivalent to p h b h , ϕ p h b h qq ď p h , f q ðñ p h , f q ď p h Ñ h , ϕ p h Ñ h qq , which is again equivalent to prove that h b h ď h ô h ď h Ñ h . This is true since b and Ñ satisfy the adjointness condition, by Lemma 4. (cid:3) We have thus proved that with residuated multilattices we obtain a residu-ated concept multilattice. Replacing one of the residuated multilattices A and A by a residuated lattice, we claim that the set of all concepts is a residuatedlattice in the following corollary. Corollarly 1.
Under the assumption of the hypothesis of Theorem 2, if A or A is a residuated lattice, then p C , ĺ q is a residuated lattice.Proof. If A is a residuated lattice the multi-infimum becomes a singleton andis indeed an infimum. Hence, every set has an infimum and so p C , ďq is acomplete residuated lower semilattice with the greatest element element p Ú , ϕ p Ú qq . The proof with A is similar. (cid:3)
5. C
ONCLUSION
By choosing residuated multilattices as set of truth values, we have provedthat the set of concepts forms a complete residuated multilattice.As next step, we will initiate and study residuated concept multilatticeswith hedges. Also, from what was done by Medina et al. [MOR07] on the useof ordered multilattices as underlying sets of truth-values for a generalizedframework of logic programming, we are planing to extend their finding in thecontext of residuated multilattices. R EFERENCES[AN72] A. Arnauld, P. Nicole,
La logique ou l’art de penser , Ãl’dition critique par Dominique De-scotes, Paris: Champion, 2011.[BV05] R. Belohlavek, and V. Vychodil,
Fuzzy attribute logic: Entailment and non-redundant ba-sis , 11th International Fuzzy Systems Association World Congress, Tsinghua, China, (2005),622 - 627.[Bm55] M. Benado,
Les ensembles partiellement ordonnés et le théorème de raffinement de Schreier.II. Théorie des multistructures , Czechoslovak Mathematical Journal, 5(3) (1955), 308-344[CCMO14] I. P. Cabrera, P. Cordero, J. Martinez, M. Ojeda-Aciego,
On residuation in multilattices:Filters, congruences, and homomorphisms , Fuzzy Sets and Systems, (2014), 1-21.[GW99] B. Ganter and R. Wille.
Formal Concept Analysis: Mathematical Foundation.
SpringerVerlag, (1999).[GJ67] J. A. Goguen,
L-fuzzy sets , Journal of Mathematical Analysis and Applications, 18(1967),145-174.[GJ68] J. A. Goguen,
The logic of inexact concepts , Synthese 19 (1968), 325-373.[HP98] P. H ´ a jek, Metamathematics of Fuzzy Logic , Kluwer, Dordrecht 1998.[HU96] U. H : o hle, On the fundamentals of fuzzy set theory , J. Math. Anal. Appl. 201 (1996), 786-826.[JT02] P. Jipsen, C. Tsinakis,
A survey of residuated lattices , In: Martà nez J. (eds) OrderedAlgebraic Structures, Developments in Mathematics, 7 (2002), Springer, Boston, MA , 19 -56.[MKLK15] L. N. Maffeu Nzoda, B. B. N. Koguep, C. Lele and L. Kwuida,
Fuzzy setting of residuatedmultilattices , Annals of fuzzy Mathematics and informatics, 10(6) (2015), 929 - 948.[MOR13] J. Medina, M. Ojeda-Aciego,and J. Ruiz-Calviño.
Concept-forming operators on multilat-tices.
Proceedings ICFCA 2013, Dresden, Germany, May 21-24, LNAI 7880, (2013), 203-215.[MOR07] J. Medina, M. Ojeda-Aciego, and J. Ruiz-Calviño.
On the ideal semantics of multilattices-based logic programs , Fuzzy Sets and Systems, 158(6) (2007), 674-688.[RM12] J. Ruiz-Calviño, and J. Medina.
Fuzzy formal concept analysis via multilattices: firstprospects and results.
Prodeedings CLA 2012, 69-79.[ZL65] L. A. Zadeh,
Fuzzy sets , Inf. Control 8, (1965) 338-353.*D
EPARTMENT OF M ATHEMATICS AND C OMPUTER SCIENCE , U
NIVERSITY OF D SCHANG , BP67, C
AMEROON
E-mail address : [email protected] **B ERN U NIVERSITY OF A PPLIED S CIENCES
E-mail address : [email protected] ***D EPARTMENT OF M ATHEMATICS AND C OMPUTER SCIENCE , U
NIVERSITY OF D SCHANG ,BP 67, C
AMEROON
E-mail address ::