The quest for rings on bipolar scales

The Quest for Ring s on Bip olar Scales Michel GRABISCH 1 , B er nard DE BAE T S 2 , and J´ anos FODOR 3 1 Universit´ e Paris I — Panth´ eon-Sorb onne LIP6, 8 rue du Capitaine Sc ott, 7501 5 Paris, F r anc e E-mail: michel.grabisch @lip6.fr 2 Dep artment of Applie d Mathematics, Biometrics and Pr o c ess Contr ol Ghent Universit y, Coupur e links 653, B- 9000 Gent , Belgium E-mail: Bernard.DeB aets@rug .ac.be 3 F aculty of V eterinary Scienc e Szent Istv´ an University, Istv´ an u. 2., H-1078 Budap est, Hungary E-mail: jfodor@univ et.hu W e consider the interv al ] − 1 , 1[ and intend to endo w it with an algebraic structure l ike a ri ng. The motiv ation lies in decision making, where scales that are symmetric w. r.t. 0 are needed in or der to represen t a kind of s ymm etry in the behaviour of the decisi on mak er. A former prop osal due to Grabisc h was based on maximum and minimum. In this pap er, we prop ose to build our structure on t-conorms and t-norms, and we relate this construction to uninorms . W e sho w that the only wa y to build a group i s to use strict t-norms, and that there is no wa y to build a ring. Lastly , we show that the main result of this paper is connected to the theory of ordered Abeli an groups. Keywor ds : ordered group, pseudo-addition, pseudo-m ultipl i cation, t-cono rm, t-norm, uninorm. 1. In tro d uctio n So far, most of the studies of aggregatio n o pe rators ha ve bee n conducted on the unit interv al [0 , 1 ], being r epresentativ e for members hip deg rees for fuzzy sets, uncertaint y degrees for probability mea sures and other non-classica l meas ures, etc. F rom psychological studies, it is k nown that human be ings ha ndle other k inds of scales. Three kinds o f scales have b een ident ified: • b ounde d unip olar s c ales : typically [0 , 1], suitable for b ounded notions such as mem ber ship deg rees and uncer taint y degrees; 1 2 The quest for rings on bip olar sc ales • unip olar sc ales (not necess arily bo unded): t ypica lly R + , suitable e.g. for pri- ority degrees (one ca n alwa ys imagine something of higher priority); • bip olar sc ales (b ounded or unbounded): t ypically R , suitable for a ll pa ired concepts o f natural lang uage, such a s attraction/ repulsion, go o d/bad, etc. In decision making, it has be en shown that the use of bip ola r s cales is o f pa rticular int erest, since they ena ble the repr esentation of symmetry phenomena in h uman behaviour, when one is face d with p ositive (gain, satisfaction, etc.) or negative (loss, dis satisfaction, etc.) scor es or utilities. This has led for example to mo dels in dec ision under uncertaint y suc h as Cumulative Pros pec t Theory 15 , based o n the symmetric Cho quet in tegral. Recently , Grabisch has in vestigated s ymmetric algebraic structures imitating the ring structure o f r eal num ber s, where the usua l + and × op eratio ns were repla ced by suitable extensions o f maximum ∨ and minimum ∧ 5 , 7 . The a im was to build an or dinal counterpart of numerical bip olar sca les, and o f the symmetric Cho quet int egral, in o rder to make some fir st steps tow a rds an ordinal version of Cumula- tive Prosp ect Theory . He prop osed new op erations, called s ymmetric maximum and minimum, and obtained a structure close to a ring . In fact, asso cia tivity was problematic, but it was nevertheless possible to define a symmetr ic Sugeno in tegral, counterpart o f the s ymmetric Cho quet integral 6 . Symmetric maximum a nd minimum can b e seen as extensions o f the usual ∨ and ∧ on [ − 1 , 1]. Re call that ∨ (re sp. ∧ ) is a t-conor m (resp. t-norm) on [0 , 1]. The question that now c omes to mind is the following: Is it p oss ible to extend any pair c onsisting o f a t-conorm a nd a t-norm to [ − 1 , 1] or ] − 1 , 1[ so that the r esulting structure is a ring ? Or, at least, is it p os sible to construct an Ab elian group? A po sitive answer to this questio n would p ermit readily to define sy mmetric fuzzy int egrals defined w ith t-conor ms and t-norms 14 . How ever, the results we present here indicate that the ques t fo r rings o n bipola r sca les seems to b e a chimeric task . 2. Bac kground In this section, we int ro duce the c oncepts needed for o ur constructions. W e refer the r eader to 11 for a comprehensive treatment. It is also the s ource of our notations. Definition 1 A triang ular nor m (t-norm for short) T is a binary op er ation on [0 , 1] such that for any x, y , z ∈ [0 , 1] the fol lowing four axioms ar e satisfie d: ( P1 ) c ommut ativity: T ( x, y ) = T ( y , x ) ; ( P2 ) asso ciativity: T ( x, T ( y , z )) = T ( T ( x, y ) , z )) ; ( P3 ) monotonicity: T ( x, y ) ≤ T ( x, z ) whenever y ≤ z ; ( P4 ) neut r al element: T (1 , x ) = x . The quest for rings on bip olar sc ales 3 An y t-norm satisfies T (0 , x ) = 0 . Typical t-norms are the minimum ( ∧ ), the alge- braic pro duct ( · ), and the Luk asiewicz t-nor m defined by T L ( x, y ) := ( x + y − 1) ∨ 0. Definition 2 A triangula r conorm (t-c onorm for short) S is a binary op er ation on [0 , 1] that, for any x, y , z ∈ [0 , 1] , satisfies P1 , P2 , P3 and ( P5 ) neut r al element: S (0 , x ) = x . An y t-cono rm satisfies S (1 , x ) = 1. Typical t-conorms are the maximum ∨ , the probabilistic sum S P ( x, y ) := x + y − xy , and the Luk asiewicz t-co norm defined b y S L ( x, y ) := ( x + y ) ∧ 1. T-norms and t-conorms are dual op erations in the sense that for a ny g iven t-nor m T , the binar y op eration S T defined by S T ( x, y ) = 1 − T (1 − x , 1 − y ) is a t-conor m (and similar ly when starting fro m S ). Hence, their prope rties are also dual. The ab ov e examples are a ll dual pair s of t-norms and t-conorms . A t-no rm (or a t-conor m) is said to b e st rictly monotone if T ( x, y ) < T ( x, z ) whenever x > 0 a nd y < z . A contin uous t-norm (resp. t-conorm) is shown to be A r chime de an if T ( x, x ) < x (resp. S ( x, x ) > x ) for all x ∈ ]0 , 1[. A strictly monotone and contin uous t-norm (resp. t-conorm) is called strict . Strict t-norms (resp. t-conor ms) ar e Archimedean. Non-strict contin uo us Archimedean t-nor ms (resp. t-conorms) a re called nilp otent . An y contin uous Archimedean t-conor m S has an additive gener ator s , i.e. a strictly increa sing function s : [0 , 1] → [0 , + ∞ ], with s (0) = 0, such that, for any x, y ∈ [0 , 1]: S ( x, y ) = s − 1 [ s (1) ∧ ( s ( x ) + s ( y ))] . (1) Similarly , any contin uous Archimedean t-norm has an additive generator t that is strictly decrea sing and sa tisfies t (1) = 0. Strict t-conor ms are characterized by s (1) = + ∞ , nilpotent t-conorms by a finite v alue of s (1 ). Additive g enerator s ar e determined up to a p ositive m ultiplica tiv e constant. If t is an a dditiv e generator of a t-norm T , then s ( x ) = t (1 − x ) is a n additive g enerator of its dual t-conor m S T . Definition 3 17 A uninorm U is a binary op er ation on [0 , 1] that, for any x, y , z ∈ [0 , 1] , satisfies P1 , P2 , P3 and ( P6 ) neut r al element: t her e exists e ∈ ]0 , 1 [ such that U ( e, x ) = x . It follows that on [0 , e ] 2 a uninorm b ehaves like a t-norm, while on [ e , 1 ] 2 it behaves lik e a t-cono rm. In the remaining parts, monotonicity implies that U is comprised betw een min a nd ma x. Asso ciativity implies that U (0 , 1 ) ∈ { 0 , 1 } . Uninorms such that U (0 , 1) = 1 ar e called disjunctive, while the others are ca lled conjunctive. By a simple resca ling, one can define from a given uninorm U a t-norm T U and a t-cono rm S U : T U ( x, y ) = 1 e U ( ex, ey ) (2) S U ( x, y ) = 1 1 − e [ U ( e + (1 − e ) x, e + (1 − e ) y ) − e ] . (3) 4 The quest for rings on bip olar sc ales Conv ers ely , fro m a given t-norm T and t-cono rm S , one can (partia lly) define a uninorm U T ,S by: U T ,S ( x, y ) = ( eT ( x e , y e ) , if ( x, y ) ∈ [0 , e ] 2 e + (1 − e ) S  x − e 1 − e , y − e 1 − e  , if ( x, y ) ∈ [ e , 1 ] 2 . (4) The rema ining parts can b e filled in with minimum or ma ximu m, leading to tw o extremal unino rms with the given underlying op er ators. The pr oblem of filling in the remaining par ts is non- trivial and has b een s olved recen tly for the ca se of a pair of contin uo us op erators 3 . Minim um and maximum, for instance, can b e extended in v ario us w ays to an idemp otent uninorm 2 . Prop ositio n 1 The fol lowing statements ar e e qu ivalent: (i) U is a u ninorm with neutr al element e , strictly monotone on ]0 , 1 [ 2 , and c on- tinuous on [0 , 1] 2 \ { (0 , 1 ) , (1 , 0) } . (ii) Ther e exists an additiv e gener ator u , i.e. a strictly incr e asing [0 , 1] → [ −∞ , ∞ ] mapping u su ch that u ( e ) = 0 and for any x, y ∈ [0 , 1] : U ( x, y ) = u − 1 ( u ( x ) + u ( y )) , (5) wher e by c onvention ∞ − ∞ = −∞ if U is c onjun ct ive, and + ∞ if U is disjunctive. Uninorms c har acterized b y the above pro po sition are called r epr esentable uninorms. Under the ab ov e assumption, it turns out that T U and S U are strict, and hav e additive g enerators t u and s u defined by: t u ( x ) = − u ( ex ) (6) s u ( x ) = u ( e + (1 − e ) x ) . (7) Definition 4 L et S b e a t- c onorm. The S -difference is define d by: x ⊖ ′ S y := inf { z | S ( y , z ) ≥ x } . (8) S -differences have b een prop ose d by W eb er 16 . F rom a logica l p oint of view, S - differences are extensions of t he binary coimplication, the logical dual of implication (in the sense of de Mor gan) 1 . S - differences a re dual to residual implicato rs o f t- norms. If S has an a dditive generator s , then it is easy to show that for any x, y ∈ [0 , 1]: x ⊖ ′ S y = s − 1 (0 ∨ ( s ( x ) − s ( y ))) . (9) If S = max, then this differ ence op er ator b ecomes: x ⊖ ′ ∨ y =  x , if x > y 0 , else . (10) In lattice theory , x ⊖ ′ ∨ y co rresp onds to the dual of the pseudo-complement of y relative to x (see e.g . 8 ). The quest for rings on bip olar sc ales 5 3. Symmetric pseudo-additions and pseudo-multiplications F ollowing o ur motiv atio n given in the int ro duction, we try to define in this section what w e mean by s ymmetric pseudo-addition and pseudo-multiplication on [ − 1 , 1 ]. Let us firs t co nsider the pseudo-addition. Our aim is to endow [ − 1 , 1] with a binar y o per ation ⊕ , ex tending a given t-c onorm S , so that we g et a co mm utative group, with neutra l element 0. A first basic fact is that we s hould conside r the ope n interv al ] − 1 , 1[, instead of the clo sed one if we want to g et a gr oup s tructure (and hence a ring). Indeed, asso ciativity implies that for any a ∈ [0 , 1[, we should hav e a ⊕ (1 ⊕ − 1) = ( a ⊕ 1) ⊕ ( − 1). Since on [0 , 1 ], ⊕ coincides with S , the a bove equalit y b ecomes a ⊕ (1 ⊕ − 1) = 1 ⊕ ( − 1), since 1 is an a bsorbing element. This equality cannot b e satisfied for all a ∈ [0 , 1[, unless 1 ⊕ ( − 1) is again an abso rbing element ; in particula r w e ca nnot hav e 1 ⊕ ( − 1) = 0 as required b y the gr oup struc ture. Hence , we exclude − 1 and 1 to get a gr oup, but we can still co nsider that ⊕ is defined o n [ − 1 , 1 ] 2 , imp osing tha t 1 ⊕ ( − 1) is equa l to 1 or − 1. As it will b e seen in Section 5, the case o f the closed int erv al cor resp onds to extende d groups . W e pro po se the following definition. Definition 5 Given a t-c onorm S , we define a symmetric pseudo-addition ⊕ as a binary op er ation on [ − 1 , 1] : ( R1 ) F or x, y ≥ 0 : x ⊕ y = S ( x, y ) . ( R2 ) F or x, y ≤ 0 : x ⊕ y = − S ( − x, − y ) . ( R3 ) F or x ∈ [0 , 1[ , y ∈ ] − 1 , 0] : x ⊕ y = x ⊖ S ( − y ) , wher e ⊖ S is a symmetrize d version of t he S -differ enc e: x ⊖ S y =        inf { z | S ( y , z ) ≥ x } , if x ≥ y − inf { z | S ( x, z ) ≥ y } , if x ≤ y 0 , if x = y for x, y ≥ 0 . Mor e over, 1 ⊕ ( − 1) = 1 or − 1 . ( R4 ) F or x ≤ 0 , y ≥ 0 : just r everse x and y . W e now give justificatio ns for o ur definition. • On [0 , 1] 2 , ⊕ has to b e commutativ e, asso ciative, ha ve neutral element 0 and absorbing element 1. Adding monoto nicit y (which may be questionable), we hav e exa ctly the ax ioms of a t-co norm. This justifies R 1 . • Since a group structure is desired, one should hav e a s ymmetric element − x for each element x ∈ ] − 1 , 1[, i.e. x ⊕ ( − x ) = 0. Using asso ciativit y and commutativit y , this leads to 0 = ( x ⊕ ( − x )) ⊕ ( y ⊕ ( − y )) = ( x ⊕ y ) ⊕ (( − x ) ⊕ ( − y )), w hic h implies ( − x ) ⊕ ( − y ) = − ( x ⊕ y ), w hic h in turn implies R2 . 6 The quest for rings on bip olar sc ales • When x ≥ 0 and y ≤ 0, w e define x ⊕ y := x ⊖ S ( − y ). Although this definition might seem to b e somewhat a rbitrary , the following consideratio ns justify this choice. F or a strict t-conorm S and x ≥ | y | , a comparison of formulas (9) a nd (1) clearly shows that they complement e ach other . F o r the case of maximum, we refer to Section 4.3. This justifies the fir st line of R3 . Notice also that in the case of a represen table unino rm U with neutral elemen t e ∈ ]0 , 1[, and a dditive gener ator u , we alwa y s have U ( x, N ( x )) = e for x ∈ ]0 , 1[, wher e N ( x ) := u − 1 ( − u ( x )) is a strong negation. After some calcula tion and appropria te r escaling (i.e. the s et [ e, 1] × [0 , e ] is linearly mapp ed to [0 , 1] × [ − 1 , 0], whence U b ecomes ⊕ a nd N ( y ) be comes − y ), one can c ome up with the formula x ⊕ y = x ⊖ ′ S ( − y ) , for ( x, y ) ∈ [0 , 1] × [ − 1 , 0 ], x ≥ − y . Co mpare this a lso with Section 4.1, es - pec ially Prop osition 2. The second line stems naturally from the requir ement − ( x ⊕ y ) = ( − x ) ⊕ ( − y ), while the third line is due to x ⊕ ( − x ) = 0. • R4 o riginates from commutativit y . The case o f ps eudo-multiplication is less problematic. W e wan t a commutativ e and as so ciative op er ation having 1 as neutral element and 0 as absorbing element , and with the same rule of sign as the pro duct, i.e. x ⊙ y = sign ( x · y ) T ( | x | , | y | ) , where the s ign function is defined a s usual by sign ( x ) :=    1 , if x > 0 0 , if x = 0 − 1 , else . This rule naturally follows from the distributivity of ⊙ w.r.t. ⊕ . Indeed, 0 = ( x ⊕ ( − x ) ) ⊙ y = ( x ⊙ y ) ⊕ (( − x ) ⊙ y ) , which e n tails ( − x ) ⊙ y = − x ⊙ y . W e pro po se the following definition. Definition 6 Given a t-n orm T , we define a symmetric ps eudo-multiplication ⊙ as a binary op er ation on [ − 1 , 1] : ( R5 ) F or x, y ≥ 0 : x ⊙ y = T ( x, y ) . ( R6 ) F or other c ases, x ⊙ y = sign ( x · y ) T ( | x | , | y | ) . Clearly , R 5 arises if we additionally require the mono tonicity of ⊙ . If the construc- tion of ⊕ and ⊙ can be done as desc rib ed ab ove, and if in addition ⊙ is distributiv e w.r.t. ⊕ , then (] − 1 , 1[ , ⊕ , ⊙ ) is a r ing. The quest for rings on bip olar sc ales 7 4. Results W e now ex amine the p ossibility o f building symmetric pseudo-additions and m ultiplications leading to a ring. W e co nsider the following four cases, cov ering all contin uous t-conor ms: 1. S is a strict t-conorm, with a dditive generator s ; 2. S is a nilp otent t-conor m, with additive g enerator s ; 3. S is the maximum op erato r; 4. S is an or dinal sum of contin uous Archimedean t-cono rms (i.e. combination of the ab ov e). 4.1. Stri ct t-c onorms Consider the symmetric ps eudo-addition ⊕ defined from a s trict t-co norm S with a dditive generato r s . Let us rescale ⊕ to a bina ry op erato r U on [0 , 1] in the following way: x ⊕ y = 2 U  1 2 x + 1 2 , 1 2 y + 1 2  − 1 , (11) and we put z := 1 2 x + 1 2 , t := 1 2 y + 1 2 (co ordinates in [0 , 1 ] 2 ). Let us int ro duce g : [ − 1 , 1] → [ −∞ , ∞ ] as follows: g ( x ) := s ( x ) for p ositive x, g ( x ) := − s ( − x ) for negative x, (12) i.e. g is a s ymmetrization of s , and is clearly a strictly increa sing function. W e int ro duce also a nother function u : [0 , 1] → [ −∞ , ∞ ] defined by u ( x ) = g (2 x − 1) that is str ictly increa sing and satisfies u ( 1 2 ) = 0. The following can b e shown (or iginally established in 10 ). Prop ositio n 2 L et S b e a strict t-c onorm with additive gener ator s , ⊕ the c orr e- sp onding pseudo-addition, and g , u define d as ab ove. Then: (i) x ⊖ y = g − 1 ( g ( x ) − g ( y )) for any x, y ∈ [0 , 1] ; (ii) x ⊕ y = g − 1 ( g ( x ) + g ( y )) for any x, y ∈ [ − 1 , 1] ; (iii) U ( z , t ) = u − 1 ( u ( z ) + u ( t )) for any z , t ∈ [0 , 1 ] . Pro of: (i) F ollows immediately from the definition of g and ⊖ . (ii) Clear ly x ⊕ y = s − 1 ( s ( x ) + s ( y )) = g − 1 ( g ( x ) + g ( y )) when x, y ∈ [0 , 1 ]. When x, y are nega tive, it holds due to R2 that x ⊕ y = − g − 1 ( g ( | x | ) + g ( | y | )) = g − 1 ( − ( g ( | x | ) + g ( | y | ))) = g − 1 ( g ( x ) − g ( y )) . 8 The quest for rings on bip olar sc ales When x is p ositive and y is negative, we write x ⊕ y = x ⊖ ( − y ) and use (i). The result then follows from the symmetry of g . (iii) Using (11), the definition of z , t and (ii), we get 2 U ( z , t ) − 1 = g − 1 ( g ( x ) + g ( y )) = g − 1 ( g (2 z − 1) + g (2 t − 1)) = g − 1 ( u ( z ) + u ( t )) . Now o bserve that for any x and y , the equality g − 1 ( y ) = x is equiv alent to y = g ( x ) = u ( 1 2 + 1 2 x ), which in turn is equiv alent to u − 1 ( y ) = 1 2 + 1 2 x = 1 2 + 1 2 g − 1 ( y ). This shows that U ( z , t ) = 1 2 g − 1 ( u ( z ) + u ( t )) + 1 2 = u − 1 ( u ( z ) + u ( t )) .  Corollary 1 Under the assumptions of Pr op osition 2, U is a uninorm that is c on- tinuous (exc ept in (0 , 1 ) and (1 , 0 ) ), is strictly incr e asing on ]0 , 1[ 2 and has neutr al element 1 2 . Mor e over, the t -norm T U induc e d is the dual of S . Pro of: Since u is strictly increa sing a nd u ( 1 2 ) = 0, U is a uninorm fo r whic h statement (ii) o f P rop osition 1 holds. Now, using (6) a nd (7) with e = 1 2 , w e obtain t ( x ) = − s ( x − 1), so that T U and S a re dual.  Now we can sta te the main result. Theorem 1 L et S b e a strict t-c onorm with additive gener ator s and ⊕ the c orr e- sp onding pseudo-addition. Then it holds that: (i) (] − 1 , 1[ , ⊕ ) is an Ab elian gr oup. (ii) Ther e exists no pseudo-multiplic ation such that (] − 1 , 1[ , ⊕ , ⊙ ) is a ring. Pro of: (i) Since ⊕ is a uninorm with neutral element 1 2 transp osed in [ − 1 , 1] 2 , commutativit y and asso cia tivity hold, and the neutral element is 0 . Mo reov er, symmetry holds s ince for a ny x ∈ ] − 1 , 1[ we can show that x ⊕ ( − x ) = 2 U  1 2 x + 1 2 , − 1 2 x + 1 2  − 1 = 2 u − 1 ( u ( z ) + u (1 − z )) = 2 u − 1 ( u ( z ) − u ( z )) − 1 = 1 using u ( z ) = − u (1 − z ), for all z ∈ [0 , 1]. (ii) By de finition of the ps eudo-multiplication, it is a sso ciative with neutral e lement 1, so only distributivity w.r.t. ⊕ remains to b e verified. If T is the t-norm asso ci- ated to ⊕ , T is distributive w.r.t. S if a nd o nly if S = max 11 . Since the maximum op erator is no t a stric t t-conorm, the construction ca nnot work.  The quest for rings on bip olar sc ales 9 4.2. Nilp otent t-c onorms F or a nilp otent t-cono rm with additive genera tor s , it is easy to see that state- men t (i) in P rop osition 2 is s till v a lid. How ever, the constr uction do es no t lea d to an asso ciative op erator , as it is easily s een in the following exa mple. Consider the Luk as iewicz t-conor m S L ( x, y ) = ( x + y ) ∧ 1 with additive generato r s L ( x ) = x . Then x ⊖ y = x − y . Let x = − 0 . 3 , y = 0 . 6 , and z = 0 . 6, then we o btain: x ⊕ ( y ⊕ z ) = − 0 . 3 ⊕ (0 . 6 ⊕ 0 . 6) = − 0 . 3 ⊕ 1 = 0 . 7 ( x ⊕ y ) ⊕ z = ( − 0 . 3 ⊕ 0 . 6) ⊕ 0 . 6 = 0 . 3 ⊕ 0 . 6 = 0 . 9 . Hence, we can even no t build a group in this case. 4.3. The maxi mum op er ator The cas e o f maximum ha s already b een studied by Gr abisch in 5 , in a genera l setting wher e the underlying universe is a symmetr ic totally ordered set, i.e. a structure L = L + ∪ L − , wher e L + is a co mplete linea r lattice with top 1 l and bo ttom O , L − := {− x | x ∈ L + } , and − x ≤ − y if and only if x ≥ y , with the conv ention − O = O . W e are interested in L + = [0 , 1 ], although subsequent results are v alid in the general case. The symmetric maximum and minimum, denoted 6 and 7 , a re defined as fo llows. a 6 b :=    − ( | a | ∨ | b | ) , if b 6 = − a and | a | ∨ | b | ∈ { − a, − b } O , if b = − a | a | ∨ | b | , else (13) a 7 b :=  − ( | a | ∧ | b | ) , if sign a 6 = sign b | a | ∧ | b | , else. (14) The structure (] − 1 , 1[ , 6 ) fails to b e a gr oup since 6 is not a lwa ys asso ciativ e. F or example, (0 . 5 6 0 . 8) 6 ( − 0 . 8) = 0 . 8 6 ( − 0 . 8 ) = 0 , which differ s fro m 0 . 5 6 (0 . 8 6 ( − 0 . 8)) = 0 . 5 6 0 = 0 . 5 . A careful study in 7 has led to the following r esult. L et us r ephrase somewhat differently our requirements for symmetric max im um and minimum: ( C1 ) 6 and 7 co incide with ∨ and ∧ o n L + , res pectively; ( C2 ) 6 and 7 ar e asso cia tive and co mm utative on L ; ( C3 ) − a is the symmetric element o f a , i.e. a 6 ( − a ) = O , for a / ∈ { 1 l , − 1 l } ; ( C4 ) − ( a 6 b ) = ( − a ) 6 ( − b ) and − ( a 7 b ) = ( − a ) 7 b , fo r any a, b ∈ L . Condition C1 corre sp onds to R1 and R5 , while conditions C3 a nd C 4 cor resp ond to R2 , R3 and R 4 . Condition C2 expresses gr oup prop erties. The fo llowing result shows tha t this ta sk is impos sible 7 . 10 The q uest for rings on bip olar sc ales Prop ositio n 3 We c onsider c onditions ( C1 ) , ( C3 ) , ( C4 ) , and denote by ( C4+ ) c ondition ( C4 ) when a and b ar e r estricte d to L + . Then: 1. Conditions ( C1 ) and ( C3 ) imply t hat asso ciativity c annot hold for 6 . 2. Under c onditions ( C1 ) and ( C4+ ) , O is neutr al for 6 . If we r e qu ir e in addition asso ciativity, then | a 6 ( − a ) | ≥ | a | . F u rther, if we r e quir e isotonicity of 6 , then | a 6 ( − a ) | = | a | . 3. Under c onditions ( C1 ) , ( C2 ) and ( C3 ) , no op er ation is asso ciative on a lar ger domain than 6 define d by (13). 4. 6 is asso ciative for any expr ession involving a 1 , . . . , a n , a i ∈ L , such that W n i =1 a i 6 = − V n i =1 a i . 5. The unique 7 satisfying ( C1 ) and ( C4 ) is given by (14), and is asso ciative. 7 is distributive w.r.t. 6 on L + and L − sep ar ately. The conflict betw een asso cia tivit y ( C2 ) and symmetry ( C3 ) is not surprising if we realize that for S = ∨ , an y element in ]0 , 1] c an b e absorbing , if combined with smaller elements. Hence the ba sic observ a tion stated in the b eginning of Sec tion 3 applies. Rescaling the sy mmetric maxim um to a binary oper ator U max on [0 , 1 ] le ads to : U max ( x, y ) :=    min( x, y ) , if x + y < 1 1 2 , if x + y = 1 max( x, y ) , if x + y > 1 . As it lacks asso ciativity , this op era tor is not a unino rm, but it is still very m uch related to it. Indeed, the op erato r U max coincides (except on the antidiagonal x + y = 1) with t wo typical idemp otent uninorms (either taking min or max o n this antidiagonal) 2 . Howev er, it ca n also b e o btained by a limit pr o cedure starting from a repres ent able uninorm U (by iteratively mo difying its additive genera tor u ) 13 . 4.4. Or dinal sums Let us start with a simple case where S is defined by a strict t-cono rm S 1 in the upper cor ner [ a, 1 ] 2 for s ome 0 < a < 1, and coincides with maximum elsewhere, that is: S ( x, y ) = ( (1 − a ) S 1  x − a 1 − a , y − a 1 − a  + a , if x, y ∈ [ a, 1] x ∨ y , else . (15) F ollowing our cons truction, we define x ⊕ y = S ( x, y ) on [0 , 1] 2 (this is R1 ). Due to R2 , R3 a nd R4 , it r emains to define x ⊖ S y , for x, y ∈ [0 , 1], x ≥ y . The following can b e shown. The quest for rings on bip olar sc ales 11 Prop ositio n 4 L et S b e define d as in (15), and s 1 b e an additive gener ator of S 1 . Then it holds that: (i) F or ( x, y ) ∈ [ a, 1] 2 , x ≥ y : x ⊖ S y = ( (1 − a ) s − 1 1  s 1  x − a 1 − a  − s 1  y − a 1 − a  + a , if x > y 0 , else ; (16) (ii) F or ( x, y ) ∈ [0 , 1] 2 \ [ a, 1] 2 , x ≥ y : x ⊖ S y = x ⊖ ′ ∨ y . (17) Pro of: (i) By definition, x ⊖ S y = inf  z ≥ a | S  y − a 1 − a , z − a 1 − a  ≥ x − a 1 − a  ∧ inf { z < a | y ∨ z ≥ x } . The second term do es not exist unless x = y , and in this ca se it is 0 . Since s 1 is strictly increa sing, the inequality in the fir st term is equiv alent to z − a 1 − a ≥ s − 1 1  s 1  x − a 1 − a  − s 1  y − a 1 − a  , and the re sult follows. (ii) Obser ve that necessarily y ≤ a . Then w e hav e x ⊖ S y = inf { z | y ∨ z ≥ x } = x ⊖ ′ ∨ y .  Note that ⊕ is discont in uous on the line x = y , even if x ≥ a . Let us now ex press ⊕ using a genera tor function, where p ossible. Prop ositio n 5 L et us define the gener ator fun ct ion g : [ − 1 , − a [ ∪ { 0 } ∪ ] a, 1] → [ −∞ , ∞ ] by: g ( x ) =      s 1  x − a 1 − a  , if x ∈ [ a, 1 ] − g ( − x ) , if x ∈ [ − 1 , − a [ 0 , if x = 0 with the c onvention g − 1 (0) = 0 . F or any x, y su ch that ( | x | , | y | ) ∈ [ a, 1] 2 , we have: x ⊕ y = g − 1 ( g ( x ) + g ( y )) . Pro of: F or y > 0, it holds that g − 1 ( y ) = x is equiv alent to y = g ( x ) = s 1 ( x − a 1 − a ), which in turn is equiv alent to s − 1 1 ( y ) = x − a 1 − a = g − 1 ( y ) − a 1 − a . Hence, we hav e g − 1 ( y ) = s − 1 1 ( y )(1 − a ) + a . Substituting in (15) and (1 6) and co nsidering sy mmetrization lead to the des ired res ult for x 6 = − y . In case x = − y , we have x ⊕ y = g − 1 (0) = 0.  12 The q uest for rings on bip olar sc ales Let us study the a sso ciativity of ⊕ . Clearly , since an additive genera tor exists, ⊕ is assoc iative when the absolute v alue of the arg ument s b elong to [ a, 1]. Let us consider the problema tic ca se ( b ⊕ c ) ⊖ S c when 0 < b < a < c < 1. W e have ( b ⊕ c ) ⊖ S c = ( b ∨ c ) ⊖ S c = c ⊖ S c = 0. How ever, b ⊕ ( c ⊖ S c ) = b . Hence asso c iativity do es not ho ld everywhere. It is easily verified that putting a seco nd strict t-conorm S 2 on [0 , a ] 2 in o rder to cov er the dia gonal doe s not change the pr oblem. Let us consider as above 0 < b < a < c < 1. Then ( b ⊕ c ) ⊖ S c = ( b ∨ c ) ⊖ S c = 0 again, and a sso ciativity is not fulfilled. W e conclude that ordinal sums ca nnot lead to asso ciative op erators , although they result in “ more asso cia tiv e” o pe rators than maximum do es. 5. On the relation with ordered Ab elian groups The result on strict t-cono rms co uld hav e be en s hown using results known in the theory of o rdered groups. W e intro duce the neces sary definitions (see e.g. 9 , 4 ). Definition 7 L et ( W, ≤ ) b e a line arly or der e d set, having top and b ottom denote d ⊤ and ⊥ , a p articular nonext r emal element e , and let u s c onsider an int ernal binary op er ation ⊕ on W , and a unary op er ation ⊖ on W such that x ≤ y if and only if ⊖ ( x ) ≥ ⊖ ( y ) : (i) ( W, ≤ , ⊕ , ⊖ , e ) is an o rdered Abelian group (O AG) if it s at isfies for al l nonex- tremal elements x, y , z : (i1) x ⊕ y = y ⊕ x ; (i2) x ⊕ ( y ⊕ z ) = ( x ⊕ y ) ⊕ z ; (i3) x ⊕ e = x ; (i4) x ⊕ ( ⊖ ( x )) = e ; (i5) x ≤ y implies x ⊕ z ≤ y ⊕ z . (ii) ( W, ≤ , ⊕ , ⊖ , e ) is an extended or dered Abelia n gr oup (OA G + ) if in addition 1. ⊤ ⊕ x = ⊤ , ⊥ ⊕ x = ⊥ for al l x , ⊖ ( ⊤ ) = ⊥ , ⊖ ( ⊥ ) = ⊤ ; 2. if x and y ar e n on-extr emal, then x ⊕ y is non-ext re mal. Clearly , our concern is to find an OA G + , with W = [ − 1 , 1], ⊤ = 1, ⊥ = − 1, ⊖ = − , and ⊕ co rresp onds to our op eration ⊕ . Definition 8 (i) An iso morphism φ of an OA G (r esp. OA G + ) W = ( W, ≤ , ⊕ , ⊖ , e ) onto an OA G ( resp . O AG + ) W ′ = ( W ′ , ≤ ′ , ⊕ ′ , ⊖ ′ , e ′ ) is a one-to- one mapping fr om W onto W ′ pr eserving the st ructur e, i.e. s u ch that (i1) φ ( x ⊕ y ) = φ ( x ) ⊕ ′ φ ( y ) ; (i2) φ ( ⊖ x ) = ⊖ ′ φ ( x ) ; (i3) φ ( e ) = e ′ ; (i4) x ≤ y if and only if φ ( x ) ≤ ′ φ ( y ) . The quest for rings on bip olar sc ales 13 (ii) W is a substructur e of W ′ if W ⊂ W ′ and the structure of W is the re stric- tion of the structur e of W ′ to W , i.e. x ⊕ y = x ⊕ ′ y , ⊖ x = ⊖ ′ x , e = e ′ , and x ≤ y if and only if x ≤ ′ y , for al l x, y ∈ W . (iii) An isomorphic em bedding of W int o W ′ is an isomorphism of W onto a substructu r e of W ′ . Observe that in (i) ab ov e, in the case of OA G + , we have in addition φ ( ⊤ ) = ⊤ ′ (due to (i1) with x = ⊤ ), a nd φ ( ⊥ ) = ⊥ ′ (due to (i2)). Definition 9 (i) An OAG W is dense if ther e is no le ast p ositive element, i.e. an element x ∈ W such that x > e , and ther e is no y ∈ W such that e < y < x . (ii) An OA G W is completely ordered if e ach non-empty b oun de d X ⊂ W has a le ast upp er b ound. Obviously , (] − 1 , 1[ , ≤ , ⊕ , − , 0) is dense and completely ordered, the sa me holds fo r the closed in terv al. Theorem 2 If W is a c ompletely or der e d and dense OAG, then it is isomorphic to ( R , ≤ , + , − , 0) . The same r esult holds if W is a n OA G + and if R is replaced by R := R ∪ {− ∞ , ∞} . This shows that ⊕ has necessarily the following form: x ⊕ y = φ − 1 [ φ ( x ) + φ ( y )] , (18) where φ : [ − 1 , 1] → R is one-to- one, o dd, incr easing, and s atisfies φ (0) = 0 . Clear ly , φ corr esp onds to the function g of Prop ositio n 2 . 6. Concluding rem arks In this pap er we ha ve studied whether it is po ssible to introduce a ring structur e on ] − 1 , 1[, using a t-co norm and a t-no rm. The results show that the answer is negative. In the pa rticular case o f the ma ximu m and minimum, the symmetric maximum a nd minimum 5 are the b est p os sible choices: they provide a str ucture that is not a ring, but is the closest p ossible to a ring in a well-defined sens e. One may be interested in finding a ring structure on finite o rdinal scales. How- ever, it is well known that there exist no stric t t-co norms on finite o rdinal scales 12 . Hence, there is no chance to o btain an Abelia n group. Nilpotent t-conorms lead to disadv antageous results as we have shown ab ov e. Hence, o ne ca n co nclude that the bes t choice seems to b e the symmetric maximum. Finally , we remar k that this structure together with the symmetric minim um is the clo sest p ossible to a r ing, due to the fact that the only chance to get distributivity is with S = max. 14 The q uest for rings on bip olar sc ales Ac knowledgemen ts This w ork is s uppo rted in par t by the Bilateral Scientific and T echnological Co- op eration Flanders–Hunga ry B IL00/5 1 (B-08/ 2000), by FKFP 00 51/20 00, and by OTKA 046 762. References 1. B. De Baets. Coimplicators, the forgotten connectives. T atr a Mt. Math. Publ. , 12:229– 240, 1997. 2. B. De Baets. Id emp otent u ninorms. Eur op e an J. Op er. R es. , 118:631– 642, 1999. 3. J. F od or, B. De Baets, and T. Calvo. Characterization of uninorms with given contin- uous underlying t- norms and t-conorms. submitted. 4. L. F uchs. Partial ly Or der e d Algebr ai c Systems . Add ison-W esley , 1963. 5. M. Grabisc h. Symmetric and asymmetric fuzzy integrals: the ordinal case. I n 6th Int. Conf. on Sof t Computing (Ii zuka 2000) , Iizuk a, Japan, Octob er 2000 . 6. M. Grabisc h. The symm et ric Sugeno integra l. F uzzy Sets and Systems , 139:473-49 0. 7. M. Grabisch. The M¨ obius function on symmetric ordered structures and its application to capacities on finite sets. Discr ete Mathematics , submitted. 8. G. Gr¨ atzer. Gener al L attic e The ory . Birkha¨ user, 2nd edition, 1998. 9. P . H´ ajek, T. Havr´ anek, and R . Jirou ˇ sek. Unc ertain Information Pr o c essing in Exp ert Systems . CRC Press, 1992. 10. E.P . Klement, R . Mesiar, and E. Pap. On the relationship of associativ e comp ensatory operators to triangular norms and conorms. I nt. J. of Unc ertainty, F uzziness and Know le dge-Base d Systems , 4:129–144, 1996. 11. E.P . Klement, R. Mesiar, and E. Pap. T riangular Norms . Kluw er Academic Publishers, 2000. 12. G. Ma yor and J. T orrens. On a class of op erators for exp ert systems. Int. J. of Intel l igent Systems , 8:771–7 78, 1993. 13. R. Mesiar and M. Komorniko v´ a. T riangular norm-based aggrega tion of ev idence un- der fuzziness. In: B. Bouc h on-Meunier, editor, A ggr e gation and F usion of Imp erfe ct Information , S tudies in F uzziness and Soft Computing, pages 11–35. Physica V erlag, 1998. 14. T. Murofushi and M. Sugeno. F uzzy t-conorm integra ls with respect to fuzzy measures : generalization of Sugeno integral and Choquet integral. F uzzy Sets and Systems , 42:57– 71, 1991 . 15. A. Tversky and D . Kahneman. Adv ances in prospect th eory: cum ulative representation of uncertain ty . J. of Risk and Unc ertainty , 5:297-323, 1992. 16. S. W eb er. ⊥ -decomp osable measures and in tegrals for arc himedean t- conorms ⊥ . J. Math. Anal. Appl. , 101:114 –138, 1984. 17. R. Y ager and A. Ryb alo v . Uninorm aggregatio n op erators. F uzzy Sets and Systems , 80:111– 120, 1996.