Bipolarization of posets and natural interpolation

Bip olarization of p osets and natural i n terp olation ∗ Mic hel GRABISCH † Universit ´ e de P ari s I – P anth ´ eon-Sorb onne Cen tr e d’Econo m ie de la Sor b onne 106-1 1 2 B d . de l’Hˆ opita l, 75 013 P a r is, F rance Christ o phe LABREUCHE Thales Resea rc h & T echnology RD 12 8, 917 67 Palaiseau C edex, F rance V ersion of August 1, 2 021 Abstract The Cho quet in tegral w.r.t. a capac ity can b e seen i n the finite case as a parsimonious linear in terp olato r b et ween v ertices of [0 , 1] n . W e take this basic fact as a starting p oin t to define the Ch o quet in tegral in a v ery general w a y , using the geometric realiza tion of lattices and th eir n atural triangulation, as in the w ork of Koshev o y . A second aim of t h e pap er is to define a general mec hanism for the bip olarizat ion of ordered structures. Bisets (or signed sets), as well as bisubmo d ular functions, bicapacities, bico op erativ e games, as w ell as the Choqu et in tegral defin ed for them can b e seen as particular in stances of this scheme. Lastly , an application to m ulticriteria aggregat ion with multiple reference lev els illustrates all the resu lts presented in the pap er. Keyw ords: in terp olatio n , Cho quet integral, lattice, bip olar stru cture ∗ A shor t a nd preliminary version of this pap er has b een presented at the RIMS Symp osium on Information and ma thematics o f nonadditivity and nonextensivity , Kyoto Univ ersity , Sept. 20 06 [13]. † Corresp o nding author. T el (+33) 1-44 -07-8 2-85 , F ax (+33) 1-4 4-07- 83-01 , email michel .grab isch@un iv-paris1.fr 1 1 In tro ductio n Capacities and the Cho quet integral [6] hav e b ecome fundamen ta l concepts in decision making (see, e.g., the w o rks of Sc hmeidler [2 2], Murofushi and Sugeno [17], and Ko shev o y [14]). An in teresting but not so well kno wn fact is that in the finite case, the Cho quet in tegral can b e obtained as a parsimonious linear interpola t io n, supp osing that v alues on the v ertices of the h yp ercub e [0 , 1] n are kno wn. The in terp olation form ula w as disco ve red b y Lo v´ asz [15], considering the problem of extending the domain of pseudo-Bo olean functions to R n . Later, Marichal [1 6 ] remark ed that this form ula was precisely the Cho quet in tegral (see also Grabisc h [8]). The idea of considering the Cho quet in tegral as a parsimonious linear interpola tor could serv e as a basic principle for extending the notion of Cho quet integral to mor e general framew orks. An ex ample of t his has b een done b y the authors in [10], considering m ultiple reference lev els in a con text of m ulticriteria aggregation. Another remark able example of generalization of the Cho quet in tegral is the one for bicapacities, prop osed b y the autho r s [11, 12]. As this pap er will mak e it clear, bicapacities are an example of concept bas ed on the of bip olarization of a partially ordered set, in this case Bo olean lattices. Sp ecifically , tak e a finite set N and the set 2 N of all its subsets ordered b y inclusion: we o btain a Bo olean lattice, and a capacit y is an isotone real-v alued mapping on 2 N . In tro ducing Q ( N ) := { ( A, B ) ∈ 2 N × 2 N | A ∩ B = ∅ } , a bicapacity is a real-v alued mapping on Q ( N ), satisfying some monotonicity condition. Observ e that Q ( N ) could b e denoted by 3 N as well: ( A, B ) ∈ Q ( N ) can b e considered as a function ξ o f {− 1 , 0 , 1 } N , where ξ ( i ) = 1 if i ∈ A , ξ ( i ) = − 1 if i ∈ B , and 0 otherwise. Then the term “ bip olarization” b ecomes clear, since 2 N ≡ { 0 , 1 } N has one “p ole” (the v alue 1, and 0 is the origin or neutral v alue), and {− 1 , 0 , 1 } N has 2 p o les, namely − 1 and 1, around the neutra l v alue 0. The set 3 N and functions defined on it are not new in the literature. T o the kn owledge of the authors, it has b een introduced a ppro ximately at the same time and indep enden tly b y Chandrasek aran and Kabadi [5], Bouchet [4], Qi [20], a nd Nak am ura [19] in the field of matroid theory and optimization, and lat er we ll dev elop ed by Ando a nd F ujishige [1]. They use the term biset or signe d sets fo r elemen ts o f 3 N , and bisubmo dular functions for bicapacities (with some more restrictions). In the field of co op erative game theory , Bilbao has introduced bic o op er ative games [2], whic h corresp onds to bicapa cities without the monotonicit y condition. Other remark able w orks on bicapacities and bico op erativ e games include the one of F ujimoto, who defined the M¨ obius tr ansform of bicapacities under the name of bip ola r M¨ obius tr ansform [7]. The aim of this paper is t w ofo ld: First to define the C ho quet in tegral in a v ery general w a y , as a parsimonious linear in terp olatio n. This is done t hrough the conce pt of geometric realization of a distributiv e lattice and its natural tria ngulation. Second, to provide a general mec hanism for the bip olarization of a p oset, and to extend the previous concepts (geometric realizat io n, Cho quet in tegral, etc.) to the bip olarized structure. Then, all concepts around bisets, bicapacities, etc., a re reco v ered as a particular case. Our w ork has b een essen tially inspired a nd motiv a t ed by Koshev o y , who use d the geometric realization of a latt ice and its natural triangulation [14], and b y F ujimoto [7], who first remark ed the inadequacy of our original definition of the M¨ obius transform for 2 bicapacities in [11 ], and prop o sed the bip olar M¨ obius transform. Section 2 intro duces necess ary material, in particular geometric realizations, natural triangulations and interpolation. Section 3 is the core sec tion of the pap er, whic h presen ts the concept of bip olarization, then the bip olar v ersion of the geometric realization, natural triangulations and interpolation. L a stly , Section 4 giv es some examples, and dev elops the particular case of the pro duct of linear lattices, whic h correspo nds to an application in m ulticriteria aggregatio n with reference lev els. W e sho w t hat results obtained previously b y the aut ho rs in [10] a r e reco v ered. 2 Preliminaries In this section, w e consider a finite index set N := { 1 , . . . , n } . 2.1 Capacities and bicapacities W e recall f rom Rota [21] that, given a lo cally finite p oset ( X , ≤ ) with b otto m elemen t, the M¨ obius f unction is the function µ : X × X → R which giv es the solution to a n y equation of the fo r m g ( x ) = X y ≤ x f ( y ) , (1) for some real-v alued functions f , g on X , b y f ( x ) = X y ≤ x µ ( y , x ) g ( y ) . (2) F unction f is called the M¨ obius tr ansform (or in v e rse ) of g . Definition 1 (i) A function ν : 2 N → R is a game if it satisfies ν ( ∅ ) = 0 . (ii) A game which satisfies ν ( A ) ≤ ν ( B ) when ever A ⊆ B (monotonicity) is c al le d a capacit y [6] or fuzzy measure [23]. The c ap acity is normalized if in addition ν ( N ) = 1 . Unanimity games are capacities of the t yp e u A ( B ) := ( 1 , if B ⊇ A 0 , else for some A ⊆ N , A 6 = ∅ . It is well kno wn t ha t the set of unanimit y games is a basis for all games, whose co ordinates in this basis are exactly the M¨ o bius transform of the game. Definition 2 L et us c onsider f : N → R + . The Cho quet in tegral of f w.r.t. a c ap acity ν is given by Z f dν := n X i =1 [ f ( π ( i )) − f ( π ( i + 1 ))] ν ( { π (1) , . . . , π ( i ) } ) , wher e π is a p ermutation on N such that f ( π (1)) ≥ · · · ≥ f ( π ( n )) , and f ( π ( n + 1)) := 0 . 3 The ab o v e definition is v alid if ν is a game. F or a ny { 0 , 1 } -v alued capacit y ν o n 2 N w e ha v e (see, e.g., [1 8]): Z f dν = _ A | ν ( A )=1 ^ i ∈ A f ( i ) . (3) The expression of the Cho quet in tegral w.r.t. the M¨ obius transform of ν (denoted by m ) is Z f dν = X A ⊆ N m ( A ) ^ i ∈ A f ( i ) . (4) W e intro duce Q ( N ) := { ( A, B ) ∈ 2 N × 2 N | A ∩ B = ∅ } . Definition 3 (i) A ma pping v : Q ( N ) → R such that v ( ∅ , ∅ ) = 0 is c al le d a bico op- erativ e game [2]. (ii) A bic o op er ative gam e v such that v ( A, B ) ≤ v ( C , D ) whenever ( A, B ) , ( C , D ) ∈ Q ( N ) with A ⊆ C and B ⊇ D (monotonicity) is c al le d a bicapacit y [9 , 11]. Mor e - over, a bic ap acity is normalize d if in addition v ( N , ∅ ) = 1 and v ( ∅ , N ) = − 1 . Definition 4 L et v b e a bic ap acity and f b e a r e al-val ue d function on N . The (g eneral) Cho quet integral of f w.r.t v is given by Z f dv := Z | f | dν N + f wher e ν N + f is a game on N define d by ν N + f ( C ) := v ( C ∩ N + f , C ∩ N − f ) , ∀ C ⊆ N and N + f := { i ∈ N | f ( i ) ≥ 0 } , N − f = N \ N + f . Note that the definition remains v alid if v is a bico op erative game. Considering on Q ( N ) the pro duct order ( A, A ′ ) ⊆ ( B , B ′ ) ⇔ A ⊆ B and A ′ ⊆ B ′ , the M¨ obius transform b of a bicapacit y v is the solution of: v ( A 1 , A 2 ) = X ( B 1 ,B 2 ) ⊆ ( A 1 ,A 2 ) b ( B 1 , B 2 ) = X B 1 ⊆ A 1 B 2 ⊆ A 2 b ( B 1 , B 2 ) . This giv es: b ( A 1 , A 2 ) = X B 1 ⊆ A 1 B 2 ⊆ A 2 ( − 1) | A 1 \ B 1 | + | A 2 \ B 2 | v ( B 1 , B 2 ) 4 (see F ujimoto [7]). Unanimity games are then natura lly defined by u ( A 1 ,A 2 ) ( B 1 , B 2 ) := ( 1 , if ( B 1 , B 2 ) ⊇ ( A 1 , A 2 ) 0 , else. and form a basis of bico op erative games. The expression of the Cho quet in tegra l in terms of b is give n b y Z f dv = X ( A 1 ,A 2 ) ∈Q ( N ) b ( A 1 , A 2 ) h ^ i ∈ A 1 f + ( i ) ∧ ^ j ∈ A 2 f − ( j ) i , (5) with f + := f ∨ 0 and f − := ( − f ) + . 2.2 Lattices, geometric realizati ons, and triangulation A lattic e is a set L endo we d with a partial order ≤ suc h that f or any x, y ∈ L their least upp er b ound x ∨ y and great est lo we r b ound x ∧ y alw a ys exist. F or finite lattices, the gr eat est elemen t of L (denoted ⊤ ) and least elemen t ⊥ a lw a ys exist. x c overs y (denoted x ≻ y ) if x > y and there is no z suc h that x > z > y . A sequence of elemen ts x ≤ y ≤ · · · ≤ z of L is called a chain from x to z , while an anticha i n is a sequence of elemen ts suc h that it con tains no pair of comparable elemen ts. A c hain from x to z is maximal if no elemen t can b e added in the chain, i.e., it has the form x ≺ y ≺ · · · ≺ z . The lattice is distributive if ∨ , ∧ ob ey distributivit y . An elemen t j ∈ L is join- irr e ducible if it cannot b e expressed as a suprem um of other elemen ts. Equiv alen tly , j is join-irreducible if it cov ers only one elemen t. Jo in-irreducible elemen ts co ve ring ⊥ are called atoms , and the lat tice is atomistic if all join- ir r educible elemen ts are a toms. The set of all jo in-irreducible elemen ts of L is denoted J ( L ). F or an y x ∈ L , w e say that x has a c omplemen t in L if there exists x ′ ∈ L suc h that x ∧ x ′ = ⊥ and x ∨ x ′ = ⊤ . The complemen t is unique if the lattice is distributiv e. An imp ortant prop ert y is that in a distributiv e lattice, an y elemen t x can b e written as a n irredundan t suprem um of join- irreducible elemen ts in a unique wa y . W e denote b y η ( x ) the (normal) de c omp osition of x , defined as the set o f join-irreducible elemen ts smaller or equal t o x , i.e., η ( x ) := { j ∈ J ( L ) | j ≤ x } . Hence x = _ j ∈ η ( x ) j (throughout the pap er, j, j ′ , . . . will alwa ys denote j oin-irreducible elemen ts). Note that this decomp osition ma y b e redundan t. W e can rephrase differently the ab o ve result in sev eral wa ys, whic h will b e useful fo r the sequel. Q ⊆ L is a downset of L if x ∈ Q , y ∈ L and y ≤ x imply y ∈ Q . F or an y subset P of L , we denote by O ( P ) the set of all do wnsets of P . Then the mapping η is an isomorphism of L on to O ( J ( L )) ( Bir khoff ’s theorem [3]). Also, η ( x ∨ y ) = η ( x ) ∪ η ( y ) , η ( x ∧ y ) = η ( x ) ∩ η ( y ) (6) if L is distributiv e. Next, downs ets o f some par t ia lly ordered set P corresp ond bijectiv ely to nonincreasing mappings from P to { 0 , 1 } . Let us denote b y D ( P ) the set of all 5 nonincreasing mappings from P to { 0 , 1 } . Then Birkhoff ’s theorem can b e rephrased as follo ws: any distributive lattic e L i s isomorphic to D ( J ( L )). Finally , note that a mapping of D ( P ) can b e considered a s a v ertex of [0 , 1] | P | . In summary , we ha v e: x ∈ L ↔ η ( x ) ∈ O ( J ( L )) ↔ 1 η ( x ) ∈ D ( J ( L )) ↔ (1 η ( x ) , 0 η ( x ) c ) ∈ [0 , 1] |J ( L ) | (7) where the notatio n (1 A , 0 A c ) denotes a v ector whose co o rdinates are 1 if in A , a nd 0 otherwise. All arro ws represen t isomorphisms, the leftmost o ne b eing an isomorphism if L is distributiv e. W e in tro duce now the notion of ge ometric r e aliza tion of a lattice, and its natural triangulation (see Koshev oy [14] for more details). F or an y pa r t ially ordered set P , w e define C ( P ) as the se t of nonincreasing mappings from P to [0 , 1]. It is a con vex p olyhedron, whose set of v ertices is D ( P ). Definition 5 The geometric realization of a distributive l a ttic e L is the set C ( J ( L )) . The natur al triangulation of C ( J ( L )) consists in partitioning C ( J ( L )) in to simplices whose vertices are in D ( J ( L )). These simplices corresp ond to maximal c hains of D ( J ( L )). The following prop osition summarizes all what w e need in the sequel. Prop osition 1 Supp ose that L is distributive, with n join-irr e ducible elements. Con sider any maximal chain C := { 1 ∅ = 0 ≺ 1 X 1 ≺ · · · ≺ 1 X |J ( L ) | = 1 } . Then (i) The s i m plex σ ( C ) is n -dimensional, an d c ontains v e rtic es (0 , . . . , 0) and (1 , . . . , 1) in [0 , 1] n . (ii) The se quenc e X 1 , . . . , X n induc es a p erm utation π : { 1 , . . . , n } → J ( L ) such that X i = { π (1) , . . . , π ( i ) } , i = 1 , . . . , n , and f ( j ) = n X i =1 α i 1 X i ( j ) = X X i ∋ j α i = n X i = π − 1 ( j ) α i , ∀ j ∈ J ( L ) . (8) Conversely, a p e rm utation π ind uc es a maximal chain if and only if it f ulfil ls the c ondition ∀ j, j ′ ∈ J ( L ) , j ≤ j ′ ⇒ π − 1 ( j ) ≤ π − 1 ( j ′ ) . (iii) The solution of (8) is α i = f ( π ( i )) − f ( π ( i + 1)) , i = 1 , . . . , n − 1 , and α n = f ( π ( n )) , (9) and α 0 = 1 − P n i =1 α i = 1 − f ( π (1)) . I n addition, f ( π (1) ) ≥ f ( π ( 2 )) ≥ · · · ≥ f ( π ( n )) . Definition 6 F or an y functional F : D ( J ( L )) → R on a distributive lattic e L , its natural extension to the geometric realization o f L is define d by: F ( f ) := p X i =0 α i F (1 X i ) for al l f ∈ in t( σ ( C )) , with C b eing a chain { 1 X 0 < 1 X 1 < · · · < 1 X p } in D ( J ( L )) , and σ ( C ) its c onvex hul l in C ( J ( L )) , with f = P p i =0 α i 1 X i . 6 The following prop osition readily follo ws from Prop osition 1 and t he ab ov e definition. Prop osition 2 L et L b e a distributive lattic e, with n join -irr e ducible el e ments, and any functional F : D ( J ( L )) → R . Consider any maximal chain C := { 1 ∅ = 0 ≺ 1 X 1 ≺ · · · ≺ 1 X |J ( L ) | = 1 } . (i) F or a ny f ∈ σ ( C ) , F ( f ) = n X i =1 [ f ( π ( i )) − f ( π ( i + 1 ))] F (1 { π (1) ,...,π ( i ) } ) (10) with f ( π ( n + 1)) := 0 . (ii) F is line ar in e ach s i m plex σ ( C ) , i.e., F ( f + g ) = F ( f ) + F ( g ) pr o vide d that f , g , f + g b el o n gs to the same σ ( C ) . Mor e over, F is line a r in F , in the sense that F + G ( f ) = F ( f ) + G ( f ) for any f . Example 1: If L is the Bo o lean lattice 2 N , with N := { 1 , . . . , n } , then J ( L ) = N (atoms). W e ha v e D ( J ( L )) = { x : N → { 0 , 1 } , x nonincreasing } , but since N is an an tichain, there is no restriction on x and D ( J ( L )) = { 0 , 1 } N , i.e., it is the set of v ertices of [0 , 1] n . Similarly , C ( J ( L )) = [0 , 1] N , whic h is the h yp ercub e itself. Consider now a maximal c hain in D ( J ( L )), denoted by C := { 1 A 0 < 1 A 1 < · · · < 1 A n } , with ∅ =: A 0 ⊂ A 1 ⊂ · · · ⊂ A n := N . It corresponds to a p erm utation π on N , with A i = { π (1) , . . . , π ( i ) } . Since J ( L ) is an antic hain, con v ersely any p ermu ta t io n corresp o nds to a maximal c hain. Using (9), we get F ( f ) = n X i =1 α i F (1 A i ) = n X i =1 [ f ( π ( i )) − f ( π ( i + 1 ))] F (1 { π (1) ,...,π ( i ) } ) , with the con v ention f ( π ( n + 1)) := 0. Putting µ ( A ) := F (1 A ), w e recognize the Cho quet integral R f dν (see Definition 2) .  This example sho ws t ha t the Cho quet in tegral is the natural extension of capacities. Hence, b y analog y , F ( f ) could b e called the Cho quet inte gr al of f w.r.t. F . Moreov er, using Remark 1, we could consider F as a game or capacit y defined ov er a sublattice of the Bo olean lat t ice 2 n . 3 Bip olar st r u ctures 3.1 Bip olar extension of L Definition 7 L et us c onsider ( L, ≤ ) an inf-sem ilattic e with b ottom element ⊥ . The bip olar extension e L of L is define d as fol lows: e L := { ( x, y ) | x, y ∈ L, x ∧ y = ⊥} , which we endow with the pr o duct or der ≤ on L 2 . 7 Remark that e L is a dow nset of L 2 . The f ollo wing holds. Prop osition 3 L et ( L, ≤ ) b e an inf-sem ilattic e. (i) ( e L, ≤ ) is an inf-semilattic e whose b ottom elemen t is ( ⊥ , ⊥ ) , wher e ≤ is the pr o duct or der on L 2 . (ii) The set of join-irr e ducible elements of e L is J ( e L ) = { ( j, ⊥ ) | j ∈ J ( L ) } ∪ { ( ⊥ , j ) | j ∈ J ( L ) } . (iii) The normal de c omp osition writes ( x, y ) = _ j ≤ x,j ∈J ( L ) ( j, ⊥ ) ∨ _ j ≤ y, j ∈J ( L ) ( ⊥ , j ) . Pro of: (i) Let us consider ( x, y ) , ( z , t ) ∈ L 2 . Then ( x, y ) ∧ ( z , t ) = ( x ∧ z , y ∧ t ) is the greatest low er b ound of ( x, y ) and ( z , t ) for the pro duct order. Supp ose x ∧ y = ⊥ and z ∧ t = ⊥ . Then ( x ∧ z ) ∧ ( y ∧ t ) = ⊥ to o, whic h prov es that the greatest low er b ound alw ay s exists in e L . (ii) clear since these are the join-irr educible elemen t of L 2 , and they a ll b elong to e L . (iii) clear fr o m (ii).  W e consider no w the M¨ obius function o v er e L . The aim is to solv e f ( x, y ) = X ( x ′ ,y ′ ) ≤ ( x,y ) , ( x ′ ,y ′ ) ∈ e L g ( x ′ , y ′ ) , ∀ ( x, y ) ∈ e L, (11) where f , g are real-v a lued functions on e L . The solution is giv en through the M¨ obius function on e L : g ( x, y ) = X ( z ,t ) ≤ ( x,y ) ( z ,t ) ∈ e L f ( z , t ) µ e L (( z , t ) , ( x, y )) . (12) The following holds. Prop osition 4 Th e M¨ obius function on e L is giv en by: µ e L (( z , t ) , ( x, y )) = µ L ( z , x ) µ L ( t, y ) . Pro of: Let us define h ( x ′ , y ) := P y ′ ≤ y x ′ ∧ y ′ = ⊥ g ( x ′ , y ′ ) for a giv en x ′ ∈ L suc h that x ′ ∧ y = ⊥ . Since y ′ ≤ y , x ′ ∧ y = ⊥ implies x ′ ∧ y ′ = ⊥ to o. Hence: h ( x ′ , y ) = X y ′ ≤ y g ( x ′ , y ′ ) . (13) By a similar argumen t, note t hat (11) can b e rewritten as f ( x, y ) = X x ′ ≤ x X y ′ ≤ y g ( x ′ , y ′ ) . (14) 8 Putting (13) in (14) giv es f ( x, y ) = X x ′ ≤ x h ( x ′ , y ) . (15) Applying M¨ obius in ve rsion to (13 ) and (15) g iv es g ( x, y ) = X t ≤ y µ L ( t, y ) h ( x, t ) , (16) for some fixed x , x ∧ y = ⊥ , a nd h ( x, y ) = X z ≤ x µ L ( z , x ) f ( z , y ) (17) for some fixed y , x ∧ y = ⊥ . Using (17) in to (16 ) leads to, fo r ( x, y ) ∈ e L : g ( x, y ) = X t ≤ y µ L ( t, y ) X z ≤ x µ L ( z , x ) f ( z , y ) = X ( z ,t ) ≤ ( x,y ) µ L ( z , x ) µ L ( t, y ) f ( z , y ) . Note that in the last equation ( z , t ) ∈ e L since z ≤ x , t ≤ y a nd x ∧ y = ⊥ imply z ∧ t = ⊥ . Comparing the ab o v e last equation with (12) giv es t he desired result.  Note that as usual, the set of functions u ( x,y ) defined b y u ( x,y ) ( z , t ) = ( 1 , if ( z , t ) ≥ ( x, y ) 0 , otherwise (18) forms a basis of the functions on e L . Theorem 1 L et L b e a finite distributive lattic e, and c ( L ) b e the set o f its c omplem e n te d elements. Then, for any x ∈ c ( L ) , its c omple ment b eing denote d by x ′ , the i n terval L ( x ) of e L define d by L ( x ) := [( ⊥ , ⊥ ) , ( x, x ′ )] and endowe d with the pr o duct or der of L 2 is isomo rphic to L , by the or der isomorph i s m φ x : L ( x ) → L , ( y , z ) 7→ y ∨ z . Th e inverse func tion φ − 1 x is given b y φ − 1 x ( w ) = ( w ∧ x, w ∧ x ′ ) . Mor e over, the jo in-irr e ducible ele ments o f L ( x ) ar e the im age of those o f L by φ − 1 x , i.e.: J ( L ( x )) = { ( j ∧ x, j ∧ x ′ ) | j ∈ J ( L ) } . Pro of: T ak e x ∈ c ( L ) a nd sho w that φ x is an order isomorphism b et w een L ( x ) and L . First remark that if y , z ∈ L , then y ∨ z ∈ L since L is a lattice. Also for an y ( y , z ) ∈ L ( x ), since y ≤ x , w e ha ve η ( y ) ⊆ η ( x ), and similarly η ( z ) ⊆ η ( x ′ ). Let us show that φ x is a bijection. Observ e that since x ∧ x ′ = ⊥ and x ∨ x ′ = ⊤ , w e ha v e η ( x ) ∩ η ( x ′ ) = ∅ and η ( x ) ∪ η ( x ′ ) = J ( L ) by (6), i.e., x and x ′ partition the 9 join-irreducible elemen ts of L . It fo llows that an y w ∈ L can b e written uniquely as w = y ∨ z , with y , z ∈ L defined b y η ( y ) = η ( w ) ∩ η ( x ) , η ( z ) = η ( w ) ∩ η ( x ′ ) . (19) Then ( y , z ) ∈ L ( x ) since η ( y ) ⊆ η ( x ) and η ( z ) ⊆ η ( x ′ ). The expression of the in v erse isomorphism φ − 1 x ( w ) = ( w ∧ x, w ∧ x ′ ) is clear from (19) and (6). T ak e ( y , z ) ≤ ( y ′ , z ′ ). This means y ≤ y ′ and z ≤ z ′ , hence y ∨ z ≤ y ′ ∨ z ′ . Con v ersely , tak e w ≤ w ′ . W e ha v e y = w ∧ x ≤ w ′ ∧ x = y ′ and similarly for z = w ∧ x ′ . Hence φ x is an order isomorphism. Finally , since φ x is an order isomorphism, the tw o lattices L and L ( x ) ha v e the same structure, and hence the same join-irreducible elemen ts.  Remark tha t in any finite lattice, ⊥ and ⊤ are complemen ted eleme nts, and L ( ⊤ ) = L , L ( ⊥ ) = L ∗ , where L ∗ is the dual of L (i.e., L with the rev erse order). An in teresting question is whether the union of all L ( x ), x ∈ c ( L ), is equal to e L . Theorem 2 L et L b e a finite distributive lattic e. Then the bip ola r extension e L c an b e written as: e L = [ x ∈ c ( L ) L ( x ) if and only if J ( L ) has al l its c onn e cte d c omp onents with a single b ottom element. Pro of: T ak e ( y , z ) ∈ e L , i.e., y , z ∈ L and η ( y ) ∩ η ( z ) = ∅ . T o find x ∈ c ( L ) suc h that ( y , z ) ∈ L ( x ) is equiv alent to satisfy the conditions (i) J ( L ) \ η ( x ) is a downse t ( x is complemen ted) (ii) η ( x ) ⊇ η ( y ), and η ( x ) ∩ η ( z ) = ∅ (( y , z ) b elongs to L ( x )). Consider J ( L ). Its Hasse dia g ram is formed of connected comp onen ts, sa y J 1 , . . . , J l . Remark that in a given connected comp onen t J k , it is not p ossible to partition it in to do wnsets. Indeed, supp ose tha t J k = D 1 ∪ D 2 , with D 1 , D 2 t w o disjoin t nonempty do wnsets. Since J k is connected, eac h x ∈ J k is comparable with another y ∈ J k . Hence, b y nonemptiness assumption, there exists x 1 ∈ D 1 whic h is comparable with some x 2 ∈ D 2 , i.e., either x 1 ≤ x 2 or the conv erse. But then x 1 ∈ D 2 (or x 2 ∈ D 1 ), whic h con tradicts the fact they are disjoin t. This prov es that complemen ted elemen ts x ∈ L are suc h that η ( x ) = ∪ k ∈ K ( x ) J k (20) for some index set K ( x ) ⊆ { 1 , . . . , l } . T ak e some ( y , z ) ∈ e L and supp ose that η ( y ) ⊆ ∪ k ∈ K ( y ) J k and η ( z ) ⊆ ∪ k ∈ K ( z ) J k . Supp ose t ha t all J k ’s hav e a single b ottom elemen t ⊥ k . Then necessarily , K ( y ) ∩ K ( z ) = ∅ , otherwise η ( y ) ∩ η ( z ) = ∅ w o uld not b e true. Then it suffices to tak e K ( x ) := K ( y ), K ( x ′ ) = { 1 , . . . , l } \ K ( x ) and the conditions (i) and (ii) ab ov e are satisfied. Con v ersely , assume that there exist some connected comp o nent J k with tw o b ot tom elemen ts, sa y ⊥ k and ⊥ ′ k . Consider y , z suc h that η ( y ) = ⊥ k and η ( z ) = ⊥ ′ k . Then ( y , z ) ∈ e L , but due to (20), no x can satisfy condition (ii) ab ov e.  10 Example 1 (ctd): Conside r L = 2 N . Then e L = Q ( N ). Since 2 N is Bo olean, an y elemen t A ⊆ N is complemen ted ( A ′ = A c ), and 2 N ( A ) = [( ∅ , ∅ ) , ( A, A c )]. Obv iously the conditions o f Theorem 2 are satisfied, th us Q ( N ) = [ A ⊆ N [( ∅ , ∅ ) , ( A, A c )] . This imp ortant result sho ws that e L is comp osed b y “tiles”, all iden tical to L (note ho we ve r that the union is not disjoin t). This suggests the follo wing definition. Definition 8 L et L b e a finite distributive lattic e, and e L its bip olar e x tension. e L is said to b e a regular mosaic i f J ( L ) has al l its c onne cte d c om p onen ts w i th a single b ottom element. There are tw o imp ortant particular cases of regular mosaics: (i) L is a pro duct of m linear lattices (totally o rdered). Then c ( L ) = { ( ⊤ A , ⊥ A c ) | A ⊆ { 1 , . . . , m }} where ( ⊤ A , ⊥ A c ) has co ordinate n um b er i equal to ⊤ i if i ∈ A , and ⊥ i otherwise. Also, ( ⊤ A , ⊥ A c ) ′ = ( ⊥ A , ⊤ A c ). This case cov ers Bo olean la ttices (case of capacities ), and lattices of the form k m , whic h w e will address in Section 4. (ii) J ( L ) has a single connected comp onen t with one b ott o m e lemen t. Then e L con tains only elemen ts of the form ( y , ⊥ ) or ( ⊥ , z ), i.e., e L = L ( ⊥ ) ∪ L ( ⊤ ). The following example sho ws a case where ˜ L is not a regular mosaic. Example 2: we consider L and J ( L ) give n on Figure 1. Ob viously , J ( L ) do es not satisfy the condition f o r pro ducing a regular mosaic, and as it can b e seen on Fig. 2, the bip olar structure cannot b e obtained as a replication of L . a b c J ( L ) ∅ abc a c ac O ( J ( L )) ≡ D ( J ( L )) Figure 1: A lattice L and the associat ed J ( L ). In grey , the complemen ted elemen ts 11 ( ∅ , ∅ ) ( abc, ∅ ) ( a, ∅ ) ( c, ∅ ) ( ac, ∅ ) ( ∅ , abc ) ( ∅ , c ) ( ∅ , a ) ( ∅ , ac ) S x ∈ c ( L ) L ( x ) ( a, c ) ( c, a ) ( ∅ , ∅ ) ( abc, ∅ ) ( a, ∅ ) ( c, ∅ ) ( ac, ∅ ) ( ∅ , abc ) ( ∅ , c ) ( ∅ , a ) ( ∅ , ac ) e L Figure 2: Left: bip olar structure computed as a replication of L . R ig h t: t he true bipo lar structure 3.2 Bip olar geometric reali zation Since e L is not a distributiv e lattice, it is not p ossible to define its geometric realization in the sense of Def. 5. Assuming that e L is a r egula r mosaic, w e pro p ose the follo wing definition. Definition 9 L et e L b e a r e g ular m o saic, and x ∈ c ( L ) . We c onsid er the mappings ξ x : J ( L ) → {− 1 , 0 , 1 } such that (i) | ξ x | is nonincr e asing (ii) ξ x ( j ) ≥ 0 if j ∈ η ( x ) (iii) ξ x ( j ) ≤ 0 if j ∈ η ( x ′ ) . The set of such func tion s is de note d by D x ( J ( L )) . Similarly, we intr o duc e C x ( J ( L )) := { f x : J ( L ) → [ − 1 , 1] such that | f x | is nonincr e a s i n g , f x ( j ) ≥ 0 if j ∈ η ( x ) , f x ( j ) ≤ 0 if j ∈ η ( x ′ ) } . (21) Then the bip olar geometric realization of L is f | L | := [ x ∈ c ( L ) C x ( J ( L )) . Prop osition 5 F or any x ∈ c ( L ) , D x ( J ( L )) is the set of vertic e s of C x ( J ( L )) . Pro of: It is plain that a n y ξ x is a v ertex of C x ( J ( L )). Conv ersely , assume f x is a v ertex suc h that f or some j ∈ J ( L ), f x ( j ) = α > 0 (or < 0). Then w e define f + ( j ) := f x ( j ) + ǫ, f − ( j ) := f x ( j ) − ǫ, and f + = f − = f x elsewhere , c ho osing 0 < ǫ < α small enough so that | f + | , | f − | remain nonincreasing. Then f + , f − b elong to C x ( J ( L )), and f x = 1 2 ( f + + f − ), whic h pro v es that f x is not a vertex .  12 Prop osition 6 L et x ∈ c ( L ) . Ther e is a bije ction ψ x : D x ( J ( L )) → L ( x ) define d b y ψ x ( ξ ) := ( y ξ , z ξ ) with η ( y ξ ) = { j ∈ J ( L ) | ξ ( j ) = 1 } , η ( z ξ ) = { j ∈ J ( L ) | ξ ( j ) = − 1 } , (22) and the inverse function is d e fine d by ψ − 1 x ( y , z ) := ξ ( y, z ) with ξ ( y, z ) ( j ) :=      1 , if j ∈ η ( y ) − 1 , if j ∈ η ( z ) 0 , otherwise , (23) for any j ∈ J ( L ) , or in mor e c omp act f o rm ξ ( y, z ) = 1 η ( y ) − 1 η ( z ) . Pro of: Since | ξ | is nonincreasing, { j ∈ J ( L ) | ξ ( j ) = 1 } and { j ∈ J ( L ) | ξ ( j ) = − 1 } are dow nsets. Hence y ξ , z ξ are w ell-defined, and b y construction ( y ξ , z ξ ) ∈ L ( x ). Let us sho w that | ξ ( y, z ) | is nonincreasing. Ass ume ξ ( y, z ) ( j ) = 1 or − 1. Then j ∈ η ( y ) ∪ η ( z ). Since these a re do wnsets, any j ′ ≤ j b elongs also to η ( y ) ∪ η ( z ). Assume ξ ( y, z ) ( j ) = 0, i.e., j 6∈ η ( y ) ∪ η ( z ). Then j ′ ≥ j cannot b elong to η ( y ) ∪ η ( z ) since they are dow nsets, hence ξ ( y, z ) ( j ′ ) = 0 . Finally , ψ x is one-to-one b ecause L is distributiv e, and so is L ( x ) (Bir khoff ’s theorem).  Example 1 (ct d): Consider L = 2 N , and some N + ⊆ N , N − := N \ N + . Then D N + ( N ) = n ξ N + : N → {− 1 , 0 , 1 } suc h that ( ξ N + ) | N + ≥ 0 , ( ξ N + ) | N − ≤ 0 o . Moreo v er, ψ N + ( ξ N + ) = ( { j ∈ N | ξ N + ( j ) = 1 } , { j ∈ N | ξ N + ( j ) = − 1 } ). Figure 3 should mak e things clear for notions in tro duced till this p oin t. Observ e that functions ξ x ∈ D x ( J ( L )) corresp onds to a subset of p oints o f [ − 1 , 1] |J ( L ) | of the form (1 A , ( − 1) B , 0 ( A ∪ B ) c ), with A ⊆ η ( x ) and B ⊆ η ( x ′ ), and that C x ( J ( L )) is the conv ex h ull of these p o in ts. W e end this section b y a ddressing the natural tria ngulation of the bip olar geomet- ric realization. Let us consider some f in C ( J ( L )), assuming f = P p i =0 α i 1 X i , with 1 X 0 , . . . , 1 X p forming a chain in D ( J ( L )). G iv en x ∈ c ( L ), let us define the corresp ond- ing f x in C x ( J ( L )) a s follows : f x := p X i =0 α i ψ − 1 x ( φ − 1 x ( η − 1 ( X i ))) = p X i =0 α i (1 X i ∩ η ( x ) − 1 X i ∩ η ( x ′ ) ) . 13 L O ( J ( L )) D ( J ( L )) ext([0 , 1] |J ( L ) | ) C ( J ( L )) L ( x ) D x ( J ( L )) ter([ − 1 , 1] |J ( L ) | ) C x ( J ( L )) φ x φ − 1 x | · | | · | | · | η 1 η conv ex hull vertices ψ x ψ − 1 x conv ex hull vertices ∈ t η ( t ) ∈ 1 η ( t ) ∈ (1 η ( t ) , 0 η ( t ) c ) f = P i α i 1 X i ∈ t = y ∨ z 1 η ( y ) ∪ η ( z ) ( y , z ) ∈ 1 η ( y ) − 1 η ( z ) ∈ (1 η ( y ) , − 1 η ( z ) , 0 ( η ( y ) ∪ η ( z )) c ) f x = P i α i (1 X i ∩ η ( x ) − 1 X i ∩ η ( x ′ ) ) y = t ∧ x, z = t ∧ x ′ 1 η ( t ) ∩ η ( x ) − 1 η ( t ) ∩ η ( x ′ ) Figure 3: Relat io ns a mong v a rious concepts intro duced. | · | indicates absolute v alue, and ter() indicates v ectors whose comp onents are − 1, 1 or 0. Explicitely , this giv es, fo r an y j ∈ J ( L ): f x ( j ) = ( P i | j ∈ X i α i , if j ∈ η ( x ) − P i | j ∈ X i α i , if j ∈ η ( x ′ ) . Hence | f x | tak es v alue 1 on X 0 , 1 − α 0 on X 1 \ X 0 , etc., and is nonincreasing. Remark that | f x | = f if f ∈ C ( J ( L )), and | f | x = f if f ∈ C x ( J ( L )). 3.3 Natural in terp olation on bip olar structures Again w e suppose that e L is a regular mosaic. As sume F : S x ∈ c ( L ) D x ( J ( L )) → R is giv en. W e w ant to define the extension F of this functional o n the bip ola r geometric realization f | L | . Let us take f ∈ f | L | = S x ∈ c ( L ) C x ( J ( L )). First, w e m ust c ho o se x ∈ c ( L ) suc h that f b elongs to C x ( J ( L )) ( x is not unique in general since in the definition o f f | L | the union is not disjoin t (see Def. 9)). Defining J ( L ) + := { j ∈ J ( L ) | f ( j ) ≥ 0 } , J ( L ) − := J ( L ) \ J ( L ) + , it suffices to take x, x ′ defined b y η ( x ) := [ k ∈ K J k , η ( x ′ ) := J ( L ) \ η ( x ) with K the smallest one suc h that J ( L ) + ⊆ S k ∈ K J k (using nota tions of pro of o f Theo- rem 2). No w, consider | f | , whic h b elongs to C ( J ( L )), and its e xpression using the natural 14 triangulation: | f | = p X i =0 α i 1 X i with 1 X 0 , . . . , 1 X p a c hain in D ( J ( L )). Then w e ha ve | f | x = f , and w e prop ose the follo wing definition. Definition 10 Assume e L is a r e gular mosaic. F or any functional F : S x ∈ c ( L ) D x ( J ( L )) → R , its natural extension to the bip o la r geometric realizatio n of e L is define d by: F ( f ) := p X i =0 α i F x (1 X i ) for al l f ∈ C x ( J ( L )) , letting | f | := P p i =0 α i 1 X i for some chain { 1 X 0 < 1 X 1 < · · · < 1 X p } in D ( J ( L )) , and F x : D ( J ( L )) → R de fi ne d by: F x (1 X i ) := F (1 X i ∩ η ( x ) − 1 X i ∩ η ( x ′ ) ) . Example (end): Let us t a k e once more L = 2 N . F or a give n f , w e define N + := { j ∈ N | f ( j ) ≥ 0 } and N − := N \ N + , w e hav e: F ( f ) = n X i =1 α i F N + (1 X i ) = n X i =1  | f ( π ( i )) | − | f ( π ( i + 1)) |  F (1 X i ∩ N + − 1 X i ∩ N − ) , where w e hav e used (9). Putting v ( A, B ) := F ( 1 A − 1 B ), w e recognize the Cho quet in tegral for bicapacities (see Definition 4).  Remark 1: Definition 10 can b e written equiv alen tly as F ( f ) = F x ( | f | ), making clear the r elat io n b etw een the functional on L and on e L . Lastly , w e address the problem of expres sing F in terms of the M¨ o bius transform of F , using Prop. 4. F or this purp ose, it is b etter to turn a giv en functional F on S x ∈ c ( L ) D x ( J ( L )) in to its equiv alen t form e F defined on e L , thanks to t he mappings ψ x , x ∈ c ( L ). D oing so, w e can use Prop. 4 and (12), and get the M¨ obius transform of e F , whic h we denote b y e m : e m ( x, y ) = X ( z ,t ) ≤ ( x,y ) ( z ,t ) ∈ e L e F ( z , t ) µ L ( z , x ) µ L ( t, y ) , ∀ ( x, y ) ∈ e L. W e need the follow ing result, whic h is a generalization of (3 ). Lemma 1 L et f ∈ C ( J ( L )) and F : D ( J ( L )) → { 0 , 1 } b eing non d e cr e asi ng and 0-1 value d. Then F ( f ) = _ T ⊆J ( L ) F (1 T )=1 ^ j ∈ T f ( j ) . 15 Pro of: ( a daptation fr om [18]) Using notations of Prop osition 1 , define i 0 ∈ J ( L ) suc h that f ( π ( i 0 )) = _ T ⊆J ( L ) F (1 T )=1 ^ j ∈ T f ( j ) . Assume for simplicit y that f ( π (1)) > f ( π (2)) > · · · > f ( π ( n )), and let us sho w that F (1 { π (1) ,...,π ( i ) } ) = ( 1 , if i ≥ i 0 0 , else. Assume i ≥ i 0 . Then for an y T ⊆ J ( L ) suc h that F (1 T ) = 1, w e ha v e f ( π ( i )) ≤ ∧ j ∈ T f ( j ). This inequality implies that T ⊆ { π (1) , . . . , π ( i ) } , and hence by monotonicity of F , we get F (1 { π (1) ,...,π ( i ) } ) = 1. No w supp ose i < i 0 . If F (1 π (1) ,...,π ( i ) ) = 1, it fo llo ws that f ( π ( i )) > f ( π ( i 0 )) ≥ ∧ i j =1 f ( π ( j )) = f ( π ( i )), a contradiction. Hence F ( 1 { π (1) ,...,π ( i ) } ) = 0. Using this r esult in (10) giv es the desired result.  The following is a generalization of (5). Prop osition 7 With the ab ove n otations, for any f ∈ f | L | and any F on S x ∈ c ( L ) D x ( J ( L )) , the fol lowing holds: F ( f ) = X ( s,t ) ∈ e L e m ( s, t ) h ^ j ∈ η ( s ) f + ( j ) ∧ ^ j ∈ η ( t ) f − ( j ) i , with f + = f ∨ 0 , f − = ( − f ) + . Pro of: T aking e F := u ( s,t ) giv en b y (18), w e ha v e b y D efinition 10 F ( f ) = F x ( | f | ) with f F x ( y ) = u ( s,t ) ( y ∧ x, y ∧ x ′ ) a nondecreasing 0- 1 v alued function, with v alue 1 iff y ∧ x ≥ s and z ∧ x ′ ≥ t . Since L is distributiv e, this conditio n writes [ η ( y ) ∩ η ( x ) ⊇ η ( s ) and η ( z ) ∩ η ( x ′ ) ⊇ η ( t )], whic h in turn is equiv alen t to [ η ( y ) ⊇ η ( s ) ∪ η ( t ) and η ( s ) ⊆ η ( x ) and η ( t ) ⊆ η ( x ′ )] since x, x ′ are complemen ted. Henc e, applying the ab ov e lemma, we get: F x ( | f | ) = _ y ≥ s ∨ t s ≤ x t ≤ x ′ ^ j ∈ η ( y ) | f ( j ) | = ^ j ∈ η ( s ) f + ( j ) ∧ ^ j ∈ η ( t ) f − ( j ) . Using linearit y of F x v ersus F x (see Prop o sition 2 (ii)) and the decomp o sition o f an y e F in the basis of f unctions u ( x,y ) , the result is pro v ed.  16 4 Applicatio n : k -ary bicapacitie s and Cho quet in te- gral This section is dedicated to the study of the lattice L := k n . W e set N := { 1 , . . . , n } . Elemen ts of L are th us v ectors in { 0 , 1 , . . . , k − 1 } n . F or commodity ( l A , l ′ − A ) denotes the elemen t t of L with t i = l if i ∈ A a nd l ′ otherwise, and w e put M := n ( k − 1). W e b egin b y giving a motiv ation of this study ro o ted in m ulticriteria decision making. 4.1 Multicriteria aggregation with reference lev els Let us consider N as the set of criteria. In the terminology of m ulticriteria decision making, an act or option is a mapping f : N → R , and f ( i ) is the sc or e o f option f on criteria i . W e ma y in tro duce reference lev els for scores, and b e in terested into the o v erall score of an option taking v alues only in the set of reference v alues (suc h options are called pur e , or p r ototypic al ). Since these options are proto t ypical, the decision mak er is able to assess their ov erall scores. The question arises then to compute the o v erall score of an option b eing not pure. Using our framew ork, there are basically t w o w ay s of answ ering this question. W e put L := k n , where k is t he nu mber of reference lev els, lab elled { 0 , 1 , . . . , k − 1 } . Observ e that join-irreducible elemen ts are of the form ( l i , 0 − i ), for an y l ∈ { 1 , . . . , k − 1 } and i ∈ N (see b elo w). The first w ay is to sa y that non-pure options b elong to a lev el only to some degree that can b e differen t fro m the complete mem b ership and the complete no n-mem b ership. Th us, as in F uzzy Set Theory [24], a membership degree is asso ciated to eac h leve l and eac h criterion, i.e., to eac h join-irreducible elemen t. F rom a kno wledge of these degrees, it is p ossible to in terp olate b etw een the v a lues known for pure options. More precisely , an o ption is an elemen t of C ( J ( L )). A degree in [0 , 1] is th us associated to all join-irr educible elemen ts. It can b e in terpreted as a mem b ership degree to the class of lev els low er or equal to the join-irreducible elemen t. Hence if an o ption b elongs at a giv en degree δ to a join- irreducible elemen t, it necessarily b elongs t o a degree great er or equal to δ to less preferred join-irreducible elemen ts. This explains wh y options shall b e non-increasing functions on J ( L ). The second wa y is to map the lattice on to a subse t of R n suc h that the P areto order on R n corresp onds precisely to the order relation on the lattice k n . F or this, w e map eac h reference lev el on R : ρ 0 < · · · < ρ k − 1 , whic h represen t the score assigned to eac h lev el. The lattice corresponds to the no des of a rectangular mesh in R n comp osed of the k reference lev els for eac h criterion. The generalized capacit y gives the v a lue asso ciated to these no des (i.e., the pure options). The non- pure options are an y p oint inside the mesh. The pro blem b ecomes th us an in terp olation problem in R n . Consider thus an option x ∈ [ ρ 0 , ρ k − 1 ] n and a generalized capacit y F : D ( J ( L )) → R . Let I ( x ) ∈ { 1 , . . . , k } n suc h that for an y i ∈ N ρ I i ( x ) − 1 ≤ x i ≤ ρ I i ( x ) . Define Φ : [ ρ 0 , ρ k − 1 ] n → [0 , 1] n as Φ i ( x ) := x i − ρ I i ( x ) − 1 ρ I i ( x ) − ρ I i ( x ) − 1 . 17 Define a capacit y v x on N b y v x ( S ) := F  1 S i ∈ N  (1 i , 0 − i ) ,..., (( I i ( x ) − 1) i , 0 − i )  ∪ S i ∈ S (( I i ( x )) i , 0 − i )  for all S ⊆ N . It cor r esp o nds to the v alue o n t he 2 n no des of the mesh just around x . One may hav e v x ( ∅ ) 6 = 0. Let v ′ x ( S ) = v x ( S ) − v x ( ∅ ) and η a p erm utation on N suc h that Φ η (1) ( x ) ≥ · · · ≥ Φ η ( n ) ( x ). v x ( ∅ ) + C v ′ x (Φ( x )) = v x ( ∅ ) + n X i =1 (Φ η ( i ) ( x ) − Φ η ( i +1) ( x )) ( v x ( { η ( 1) , . . . , η ( i ) } ) − v x ( ∅ )) = v x ( ∅ ) (1 − Φ η (1) ( x )) + n X i =1 (Φ η ( i ) ( x ) − Φ η ( i +1) ( x )) v x ( { η ( 1) , . . . , η ( i ) } ) . (24) 4.2 The unip olar case The set J ( L ) of join-ir r educible elemen ts is J ( L ) = n ( l i , 0 − i ) | l ∈ { 1 , . . . , k − 1 } , i ∈ N o . It is a p oset with n connected comp onents, each of them b eing the linear la ttice { 1 , . . . k − 1 } . Let us consider f an elemen t of C ( J ( L )). W e set for commo dity f l i := f ( l i , 0 − i ). The na tural triangulation of C ( J ( L )) is done throug h chains in D ( J ( L )), and maximal c hains corresp ond to some p erm utations on J ( L ) (see Prop osition 1). F or commo dity to eac h p erm utatio n π : { 1 , . . . , M } → J ( L ) w e assign t wo functions λ : { 1 , . . . , M } → { 1 , . . . , k − 1 } a nd θ : { 1 , . . . , M } → { 1 , . . . , n } suc h t hat π ( i ) = ( λ ( i ) θ ( i ) , 0 − θ ( i ) ), for all i ∈ { 1 , . . . , M } . Applying Proposition 1 aga in, w e kno w that f o r any elemen t f of a simplex of C ( J ( L )) corresp onding to a p erm utation π on J ( L ), w e ha v e f λ (1) θ (1) ≥ f λ (2) θ (2) ≥ · · · ≥ f λ ( M ) θ ( M ) and f ( l p , 0 − p ) = X i ∈{ 1 ,...,M } α i 1 X i ( l p , 0 − p ) where X i := { ( λ (1) θ (1) , 0 − θ (1) ) , . . . , ( λ ( i ) θ ( i ) , 0 − θ ( i ) ) } , α i = f λ ( i ) θ ( i ) − f λ ( i +1) θ ( i +1) for i ∈ { 1 , . . . , M − 1 } , and α M = f λ ( M ) θ ( M ) . A k -ary c ap acity is a function F : D ( J ( L )) → R . Applying Prop osition 2 the natura l extension of f ∈ C ( J ( L )) w.r.t. F is F ( f ) = M X i =1 h f λ ( i ) θ ( i ) − f λ ( i +1) θ ( i +1) i × F  1 { ( λ (1) θ (1) , 0 − θ (1) ) ,..., ( λ ( i ) θ ( i ) , 0 − θ ( i ) ) }  , with f λ ( M + 1) θ ( M +1) := 0. This could b e considered as the Cho quet integral of f w.r.t F . 18 T o reco v er t he in terp o lation f o rm ula (24) of Section 4.1, w e consider a particular class of elemen ts f in C ( J ( L )) satisfying for all i ∈ N f 1 i = · · · = f J i ( f ) − 1 i = 1 f J i ( f ) i = z i f J i ( f )+1 i = · · · = f k − 1 i = 0 , for some giv en integers J 1 ( f ) , . . . , J n ( f ) in { 1 , . . . , k − 1 } , and real n umbers z 1 , . . . , z n ∈ [0 , 1]. Let us denote b y σ a p erm utatio n on N such that z σ (1) ≥ · · · ≥ z σ ( n ) . Remark that f b elongs to all M - dimensional simplices of C ( J ( L )) whose corresp onding p erm utation satisfy: ∀ i ∈ { 1 , . . . , q f } , f λ ( i ) θ ( i ) = 1 ∀ i ∈ { q f + 1 , . . . , q f + n } , f λ ( i ) θ ( i ) = z σ ( i − q f ) ∀ i ∈ { q f + n + 1 , . . . M } , f λ ( i ) θ ( i ) = 0 where q f = P i ∈ N ( J i ( f ) − 1). Hence, f b elongs to the interior of a n -dimensional simplex corresp onding to the c hain 1 X q f < 1 X q f ∪{ (( J σ (1) ( f )) σ (1) , 0 − σ (1) ) } < · · · < 1 X q f ∪{ (( J σ (1) ( f )) σ (1) , 0 − σ (1) ) ,..., (( J σ ( n ) ( f )) σ ( n ) , 0 − σ ( n ) ) } , with X q f := { ( l i , 0 − i ) | 1 ≤ l i ≤ J i ( f ) − 1 } . Then F ( f ) = (1 − z σ (1) ) F (1 X q f )+ n X i =1 ( z σ ( i ) − z σ ( i +1) ) F (1 X q f ∪{ (( J σ (1) ( f )) σ (1) , 0 − σ (1) ) ,..., (( J σ ( i ) ( f )) σ ( i ) , 0 − σ ( i ) ) } ) (25) with z σ ( n +1) := 0. Let x ∈ [ ρ 0 , ρ k − 1 ] n defined b y x i := ρ J i ( f ) − 1 + ( ρ J i ( f ) − ρ J i ( f ) − 1 ) × z i . Then expression (24) and (25) lead to exactly the same v alue since J ( f ) = I ( x ), σ = η , z = Φ( x ) and v x ( S ) := F (1 X q f ∪ S i ∈ S (( J i ( f )) i , 0 − i ) ) . 4.3 The bip olar case The bip ola rization of L is ˜ L = n ( x, y ) ∈ k n × k n | ∀ i ∈ N , x i 6 = 0 ⇒ y i = 0 , and y i 6 = 0 ⇒ x i = 0 o . Moreo v er, the set of complemen ted elemen ts is c ( L ) = n (( k − 1) A , 0 − A ) | A ⊆ N o , 19 and (( k − 1 ) A , 0 − A ) ′ = (0 A , ( k − 1) − A ). Note that ˜ L is a regular mosaic, hence Theorem 2 applies and ˜ L is the union of all L ( x ), with x ∈ c ( L ), and L (( k − 1) A , 0 − A ) = n  ( x A , 0 − A ) , (0 A , y − A )  | x ∈ { 0 , . . . , k − 1 } | A | , y ∈ { 0 , . . . , k − 1 } n −| A | o . Let f ∈ C (( k − 1) A , 0 − A ) ( J ( L )), and f l i := f ( l i , 0 − i ). W e hav e f l i ≥ 0 if i ∈ A and f l i ≤ 0 if i 6∈ A . W e consider a simplex of C (( k − 1) A , 0 − A ) ( J ( L )) con taining f , whose corresp o nding p er- m utation is π : { 1 , . . . , M } → J ( L ), and w e define as in Section 4.2 the functions λ : { 1 , . . . , M } → { 1 , . . . , k − 1 } , a nd θ : { 1 , . . . , M } → { 1 , . . . , n } . Then    f λ (1) θ (1)    ≥    f λ (2) θ (2)    ≥ · · · ≥    f λ ( M ) θ ( M )    and   f ( l p , 0 − p )   = X i ∈{ 1 ,...,M } α i 1 X i ( l p , 0 − p ) where X i := { ( λ (1) θ (1) , 0 − θ (1) ) , . . . , ( λ ( i ) θ ( i ) , 0 − θ ( i ) ) } , α i =    f λ ( i ) θ ( i )    −    f λ ( i +1) θ ( i +1)    for i ∈ { 1 , . . . , M − 1 } , and α M =    f λ ( M ) θ ( M )    . A k -a ry bic ap acity is a f unction F : ∪ A ⊆ N D (( k − 1) A , 0 − A ) ( J ( L )) → R . The natural extension F ( f ) is: F ( f ) = M X i =1     f λ ( i ) θ ( i )    −    f λ ( i +1) θ ( i +1)     × F  1 S q ∈{ 1 ,...,i } θ ( q ) ∈ A ( λ ( q ) θ ( q ) , 0 − θ ( q ) ) − 1 S q ∈{ 1 ,...,i } θ ( q ) 6∈ A ( λ ( q ) θ ( q ) , 0 − θ ( q ) )  , with f λ ( M + 1) θ ( M +1) := 0. As b efore, this could b e considered as the Cho quet in tegra l o f f w.r.t. F . W e consider no w a particular class of elemen ts f in C ( J ( L )) satisfying for all i ∈ N | f 1 i | = · · · = | f J i ( f ) − 1 i | = 1 | f J i ( f ) i | = z i | f J i ( f )+1 i | = · · · = | f k − 1 i | = 0 , for some giv en integers J 1 ( f ) , . . . , J n ( f ) in { 1 , . . . , k − 1 } , and real n umbers z 1 , . . . , z n ∈ [0 , 1]. Let us denote b y σ a p erm utatio n on N such that z σ (1) ≥ · · · ≥ z σ ( n ) . Remark that f b elongs to all M - dimensional simplices of C ( J ( L )) whose corresp onding p erm utation satisfy: ∀ i ∈ { 1 , . . . , q f } , f λ ( i ) θ ( i ) = 1 if θ ( i ) ∈ A, and − 1 otherwise ∀ i ∈ { q f + 1 , . . . , q f + n } , f λ ( i ) θ ( i ) = z σ ( i − q f ) if θ ( i ) ∈ A, and − z σ ( i − q f ) otherwise ∀ i ∈ { q f + n + 1 , . . . M } , f λ ( i ) θ ( i ) = 0 20 where q f = P i ∈ N ( J i ( f ) − 1) . Then F ( f ) = (1 − z σ (1) ) V ( ∅ ) + n X i =1 ( z σ ( i ) − z σ ( i +1) ) V ( { σ (1) , . . . , σ ( i ) } ) , (26) with z σ ( n +1) := 0, and V ( S ) := F  1  X q f ∪ S i ∈ S (( J i ( f )) i , 0 − i )  ∩J ( L ) + − 1  X q f ∪ S i ∈ S (( J i ( f )) i , 0 − i )  ∩J ( L ) −  , with X q f := { ( l i , 0 − i ) | 1 ≤ l i ≤ J i ( f ) − 1 } . The p ositiv e part of the scale is represen ted b y the p o sitive lev els ρ 0 , . . . ρ k − 1 . The negativ e part of the scale is represen ted by the negativ e lev els ρ − k +1 , . . . , ρ 0 . Hence ρ 0 = 0 is the neutral elemen t demarcarting b et w een att ractiv e and repulsiv e v alues. Let x ∈ [ ρ − k +1 , ρ k − 1 ] n defined b y x i :=  ρ J i ( f ) − 1 + ( ρ J i ( f ) − ρ J i ( f ) − 1 ) × z i if i ∈ A ρ − J i ( f )+1 + ( ρ − J i ( f ) − ρ − J i ( f )+1 ) × z i if i 6∈ A Then pro ceeding as in Section 4.1, it is easy t o see that (26) corresponds exactly to the Cho quet integral for k - ary bicapacities defined in [10]. 4.4 Example W e end this section by illustrating the ab o ve results with k = 3 and n = 2. Clearly , the case k = 2 w as already w ell described (capacities and bicapacities). Elemen ts in L := 3 2 are denoted b y pa irs ( l 1 , l 2 ), with l i ∈ { 0 , 1 , 2 } , i = 1 , 2. W e ha v e four join-irreducible elemen ts (1 , 0) , (2 , 0) , (0 , 1) , (0 , 2). Let us consider the following function f in C ( J ( L )): f (1 , 0) = 0 . 5 , f (2 , 0) = 0 . 1 , f (0 , 1) = 0 . 3 , f (0 , 2) = 0 . 2 . Note that f is indeed nonincreasing on J ( L ). The asso ciated p ermu ta t io n is π (1) = (1 , 0) , π (2) = ( 0 , 1) , π (3) = (0 , 2) , π (4) = (2 , 0) , and the corresp onding maximal c hain is (expressed in L for simplicit y) (0 , 0) < (1 , 0) < (1 , 1) < (1 , 2) < (2 , 2) . (see Fig. 4 for a diagram of L , and the maximal c hain in b old) Supp osing F b eing defined on L , the Cho quet in tegral of f w.r.t. F is giv en by F ( f ) = [ f (1 , 0) − f (0 , 1)] F (1 , 0) + [ f (0 , 1) − f (0 , 2)] F (1 , 1) + [ f (0 , 2) − f (2 , 0)] F (1 , 2 ) + f (2 , 0) F (2 , 2) . Let us turn to the bip olar case. T o a void a hea vy notation, elemen ts of ˜ L are denoted b y ( ij, k l ) instead o f (( i, j ) , ( k , l )). The set of complemen ted elemen ts together with their 21 complemen ted elemen ts is A = ∅ : (0 , 0) ↔ (2 , 2) A = { 1 } : (2 , 0) ↔ (0 , 2) A = { 2 } : (0 , 2) ↔ (2 , 0) A = { 1 , 2 } : (2 , 2) ↔ (0 , 0) Then ˜ L = L (0 , 0) ∪ L (2 , 0) ∪ L (0 , 2) ∪ L (2 , 2), with L (0 , 0) = { (0 0 , k l ) | k , l ∈ { 0 , 1 , 2 }} L (2 , 0) = { ( i 0 , 0 l ) | i, l ∈ { 0 , 1 , 2 }} L (0 , 2) = { (0 j, k 0) | j, k ∈ { 0 , 1 , 2 }} L (2 , 2) = { ( ij, 00) | i, j ∈ { 0 , 1 , 2 }} . Consider the function f defined by f (1 , 0) = 0 . 5 , f (2 , 0) = 0 . 1 , f (0 , 1) = − 0 . 3 , f (0 , 2) = − 0 . 2 . Then A = { 1 } , f ∈ C (2 , 0) ( J ( L )), and the p erm utation π is the same as a b ov e. No w, assuming F defined on ˜ L is giv en, F ( f ) = [ | f (1 , 0) | − | f (0 , 1) | ] F (1 0 , 00) + [ | f (0 , 1) | − | f (0 , 2) | ] F (10 , 01) + [ | f (0 , 2) | − | f ( 2 , 0) | ] F (10 , 02) + | f (2 , 0) | F (2 0 , 02) . Fig. 4 sho ws the bipo lar extension ˜ L , the part L (2 , 0 ) used for f is in grey , and the maximal c hain is in b old. (1 , 1) (1 , 0) (0 , 0) (0 , 1) (2 , 0) (0 , 2) (2 , 1) (1 , 2) (2 , 2) L = 3 2 (00 , 22) (00 , 20) (02 , 20) (02 , 00) (22 , 00) (20 , 00) (20 , 02) (00 , 02) (00 , 00) (10 , 01) (10 , 00) (20 , 01) (10 , 02) (00 , 01) L = e 3 2 Figure 4: The lattice L = 3 2 and its bip olar extension. In b o ld the maximal chain corresp onding to f 5 Conclud ing remarks W e hav e provide d a g eneral sc heme fo r the bip o larization of a class of p osets, precisely of inf-semilattices. The bip olarizatio n is par ticularly simple in t he case o f a finite distributiv e 22 lattice L , where all connected comp onen ts of the p o set of join-irreducible elemen ts ha v e a least elemen t (this is t he case, e.g., for Bo olean lattices and pro ducts of linear lattices). 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