The Domination Relation Between Continuous T-Norms

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1 The Domination Relation Between Continuous T-Norms Susanne Saminger Department of Knowledge-Based Mathematical Systems, Johannes Kepler University Linz, Altenbergerstrasse 69, A-4040 Linz, Austria Bernard De Baets Department of Applied Mathematics, Biometrics, and Process Control, Ghent University, Coupure links 653, B-9000 Gent, Belgium Hans De Meyer Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 8, B-9000 Gent, Belgium Abstract The present contribution deals with domination in the framework of continuous t-norms. Basic properties and recent results for continuous Archimedean and continuous ordinal sum t-norms are presented and discussed. The domination property within several families of t-norms is mentioned. Keywords: Domination, triangular norms. Introduction The property of domination has been introduced in the framework of probabilistic metric spaces as a relation on the class of all triangle functions [], soon generalized to operations on a partially ordered set []. Domination plays an important role in constructing Cartesian products of probabilistic metric spaces, but also for the preservation of several properties, most of them expressed by some inequality, during aggregation processes (see, e.g., [3, 4, 5]). Therefore, the concept of domination has been generalized to and studied in the framework of aggregation operators [3]. Investigating domination in the framework of t- norms, it can be certified quite easily that domination is a reflexive and antisymmetric property. The crucial but still open question is whether it is also transitive (see, e.g., []). By this contribution, the authors intend to provide an overview of and insight into important and new results, to recent advances about domination in the framework of continuous t-norms. 47 Preliminaries. Triangular norms Definition ([, 6]). A t-norm T is an increasing, associative, commutative binary operation on [0, ] with neutral element. Well-known examples of t-norms are the minimum T M, the product T P, the Lukasiewicz t-norm T L (x, y) = max(x+y, 0) and the drastic product { 0, if (x, y) [0, [, T D (x, y) = min(x, y), otherwise. Definition. Let (]a α, b α [) α I be a system of pairwise disjoint open subintervals of [0, ] and let (T α ) α I be a family of t-norms. Then the ordinal sum T = ( a α, b α, T α ) α I : [0, ] [0, ] is given by a α + (b α a α)t α( x aα b α a α, y aα b α a α ), T (x, y) = if (x, y) [a α, b α], min(x, y), otherwise and is again a t-norm. For any t-norm T, its isomorphic transform T ϕ : [0, ] [0, ] w.r.t. some order isomorphism ϕ: [0, ] [0, ] is defined by T ϕ (x, y) = ϕ (T (ϕ(x), ϕ(y))) and is again a t-norm. The class of all continuous t-norms is characterized in the following way: Proposition 3 ([6, 7]). A function T : [0, ] [0, ] is a continuous t-norm if and only if it is uniquely representable as an ordinal sum of continuous Archimedean t-norms, i.e., T = ( a α, b α, T α ) α I for some index set I.

2 . Property of domination Definition 4. Consider two t-norms T, T. We say that T dominates T, T T, if for all x, y, u, v [0, ] T (T (x, y), T (u, v)) T (T (x, u), T (y, v)) () Note that due to the fact that is the common neutral element of all t-norms, domination of two t-norms implies their order (see also [5]). The converse does not hold, not even for strict t-norms (see, e.g., [6]). Similarly to the ordering of t- norms, any t-norm T is dominated by itself and by T M, and moreover dominates T D. Further, domination of two t-norms is preserved (resp. reversed) if the involved t-norms are transformed by some order-preserving (resp. orderreversing) bijection. Proposition 5 ([3]). Consider two t-norms T and T. Then T dominates T if and only if for all order isomorphisms ϕ: [0, ] [0, ] it holds that (T ) ϕ dominates (T ) ϕ. Taking into account the results of Proposition 3 we will now focus on continuous Archimedean t- norms and will then turn to continuous ordinal sum t-norms. 3 Continuous Archimedean t-norms It is well known (see, e.g., [6]) that continuous Archimedean t-norms T are exactly those t-norms having a continuous additive generator, i.e., there exists a continuous, strictly decreasing function t: [0, ] [0, ] with t() = 0 which is uniquely determined up to a positive multiplicative constant, such that for all x, y [0, ] it holds that T (x, y) = t (min(t(0), t(x) + t(y))). Moreover, the class of all continuous Archimedean t-norms is partitioned into two parts the class of all strict t-norms, all of them being isomorphic to the product T P, and the class of all nilpotent t-norms, all of them being isomorphic to the Lukasiewicz t-norm T L (see also [6]). 3. Isomorphic transformations Suppose that the problem under investigation is whether T dominates some T with T, T being two continuous Archimedean t-norms. Since domination is preserved during transformations by order isomorphisms, this problem can be transformed into one of the following cases. If T is nilpotent and therefore possesses some zero divisor, necessarily also T has to be nilpotent. Moreover, there exist some order isomorphisms ϕ, ψ : [0, ] [0, ] such that (T ) ϕ = T L and (T ) ψ = T L leading to T T (T ) ψ T L T L (T ) ϕ. If T is strict, T can be either strict or nilpotent. In both cases there exist order isomorphisms ϕ, ψ : [0, ] [0, ] such that the problem of domination can be again equivalently formulated as T T (T ) ψ T L T P (T ) ϕ in case T is nilpotent, and if T is strict as T T (T ) ψ T P T P (T ) ϕ. Summarizing, it suffices to investigate the classes of t-norms dominating or being dominated either by T P or by T L. In the framework of aggregation operators, it has already been shown that there exists a close relationship between generated t-norms and subadditive aggregation operators which we briefly recall here for the reader s convenience. Theorem 6 ([3, 8]). Consider a continuous Archimedean t-norm T with additive generator t : [0, ] [0, ], t (0) = c. A t-norm T dominates T if and only if there exists a binary subadditive aggregation operator H : [0, c] [0, c] such that T (x, y) = t (H(t (x), t (y))) for all x, y [0, ]. Note that the binary aggregation operator H denotes a binary operation on the interval [0, c] which is increasing in each component and fulfils H(0, 0) = 0 and H(c, c) = c. Moreover, it is called subadditive on [0, c] if for all x, y, u, v [0, c] with x + y [0, c] and u + v [0, c] it holds that H(x + y, u + v) H(x, u) + H(y, v). Which conclusions can be drawn from this result for t-norms dominating, e.g., T L? Proposition 7. If a t-norm T dominates T L, then it is -Lipschitz, i.e., a -copula. 48

3 Note that being a -copula is a necessary (but not sufficient) condition for a t-norm T to dominate T L, but not all -copulas dominate T L. Consider, e.g., the -copula C(x, y) = max(0, min(x, y 3, 3 )) + max(0, min(x 3, y, 3 )) + max(0, min(x 3, y 3, 3 )). Then we get for x = u = v = y = 3 0 = C( 3, 3 ) = C(T L( 3, 3 ), T L( 3, 3 )) < T L (C( 3, 3 ), C( 3, 3 )) = T L( 3, 3 ) = 3 and therefore C does not dominate T L. Note that an Archimedean t-norm is a -copula if and only if its additive generator is convex []. 3. Investigation of additive generators Since all continuous Archimedean t-norms possess continuous additive generators, their influence on the property of domination has been studied. The first result deals with powers of additive generators. Proposition 8 ([6]). Consider a continuous Archimedean t-norm T with additive generator t. For each ]0, [ the function t : [0, ] [0, ], x (t(x)) is an additive generator of a continuous Archimedean t-norm denoted by T (). For, µ ]0, [ with > µ it holds that T () T (µ). Note that this proposition provides not only insight into the structure of domination but also an idea how to construct dominating t-norms. Moreover, it can be directly applied to prove domination within several families of t-norms. Example 9. Consider the family of Aczél-Alsina (T AA ) [0, ], for ]0, [ they are generated by t AA (x) = ( log(x)), clearly being powers of the additive generator of T P. Since T0 AA = T D and T AA = T M, which are dominated by resp. dominate all t-norms, it holds that for all, µ [0, ] with > µ it follows that T AA Tµ AA. Note that the members of the Aczél-Alsina family are strict for ]0, [. Example 0. The members of the Dombi family (T D ) [0, ] are for ]0, [ generated by 49 t D x (x) = ( x ) being powers of the additive generator of the Hamacher product. Once again, T0 D = T D and T D = T0 H such that for all, µ [0, ] with > µ it follows T D T µ D. Note that the members of the Dombi family are strict for ]0, [. Example. Consider the family of Yager t- norms (T Y) [0, ], for ]0, [ generated by the additive generators t Y (x) = ( x). One sees immediately that these additive generators are just powers of the additive generator of T L. Moreover, since T0 Y = T D and T Y = T M, for all, µ [0, ] with > µ, it holds that T Y T µ Y. Note that the members of the Yager family are nilpotent for ]0, [. The fact that continuous Archimedean t-norms are generated by some continuous additive generator can be even applied directly to the inequality of domination leading to the following result. Proposition. Consider two continuous Archimedean t-norms T, T with additive generators t, t. Then T dominates T if and only if h: [0, ] [0, ], x t t ( ) (x) fulfills the following inequality for all a, b, c, d [0, t (0)] h ( ) (h(a) + h(c)) + h ( ) (h(b) + h(d)) h ( ) (h(a + b) + h(c + d)) () Note that h ( ) = t t ( ) since the t-norms are continuous. Due to the induced order on the t-norms, necessarily t t : [0, t (0)] [0, ] has to be superadditive on [0, t (0)] (see also [6]). As a consequence, in case that T is nilpotent, h: [0, ] [0, ] has to be superadditive on [0, t (0)] and is constant for all x t (0), i.e., h(x) = t (0) for all x [t (0), ]. If T and T are two strict t-norms, h is a function on [0, ], moreover, () has to be fulfilled for all a, b, c, d [0, ] and is often referred to as the Mulholland inequality [9], being a generalization of the Minkowski inequality (see, e.g., [0]). This special case, h being a strictly increasing, continuous function on [0, ], has been treated extensively in the literature (see, e.g., [9, 0,, ]) mainly focussing on necessary and sometimes also

4 on sufficient conditions on h to fulfill the Mulholland Inequality. We just briefly recall two of them due to Mulholland and Tardiff, since although all the results might be of mathematical interest, domination within subclasses of t-norms most often has been proven or disproven by other methods. However, remarkable is that, also in this case, the close relationship between the domination property and convex or superadditive functions can be recognized, properties which can be easily visualized by corresponding plots. Proposition 3 ([0]). Let h: [0, [ [0, [ be a continuous, strictly increasing function fulfilling h(0) = 0. If h and log h exp are convex, then h fulfills h (h(a) + h(c)) + h (h(b) + h(d)) for all a, b, c, d [0, [. h (h(a + b) + h(c + d)) (3) Proposition 4 ([]). Let h: [0, [ [0, [ be a continuous, strictly increasing function fulfilling h(0) = 0. If h fulfills (3), then it is convex and thus superadditive. 3.3 Short remark on further families We have already introduced results for some families of t-norms and, for the sake of completeness, we would like to add some more. Remarkably and as already mentioned before, the proofs for these families although they contain subclasses of strict t-norms have not been achieved by applying the conditions for the Mulholland inequality but directly through some other techniques. In [3], Sherwood showed that in the Schweizer-Sklar family of t-norms (T SS) [, ] domination is reversely related to the parameter of the family, more precisely T SS Tµ SS if and only if µ. Note that the members of the Schweizer- Sklar family are strict for ], 0] and nilpotent for ]0, [. Sarkoci [4] has proven that the only possible cases of domination within the Frank family of t-norms (T F ) [0, ] are that for each t- norm T F only T F T F, T F 0 = T M T F and T F T F = T L hold. Note that all Frank t-norms with ]0, [ are strict. Moreover, since the rare occurrence of domination in this family, transitivity of domination within this family is fulfilled, expressing that is a (very sparse) partial order relation on the Frank family of t-norms. Sarkoci [4] has also shown that the same situation applies for the Hamacher family of t-norms (T H) [0, ], which are strict for ]0, [. For each member T H of the Hamacher family it just holds that T H T H, T H T H = T D, and T0 H T H. 4 Continuous ordinal sums Remember that an ordinal sum is continuous if and only if all its summands are continuous. When discussing domination among ordinal sum t-norms, we have to take into account the underlying structure of the ordinal sum. In case both involved ordinal sum t-norms are determined by the same family of non-empty, pairwise disjoint open subintervals, domination of the ordinal sums is based on the domination of all corresponding summand t-norms. Proposition 5. Let (]a α, b α [) α I be a family of non-empty, pairwise disjoint open subintervals of [0, ]. Further consider two families (T,α ) α I, (T,α ) α I of t-norms and build the corresponding ordinal sums T = ( a α, b α, T,α ) α I and T = ( a α, b α, T,α ) α I. Then T dominates T if and only if T,α dominates T,α for all α I. What if the structure of both ordinal sums is not the same? For these cases we provide some necessary conditions based on the order and on idempotent elements illustrating the problem of domination in this framework. Let us therefore assume that the involved ordinal sums are based on two at least partially different structures and families, i.e., T = ( a α, b α, T α ) α I and T = ( a β, b β, T β ) β J. Since for a continuous t-norm T the existence of a non-trivial idempotent element d is equivalent to being representable as an ordinal sum T = ( 0, d, T, d,, T ) of some t-norms T and T (see also [6]), we assume w.l.o.g. that the previous representations of T and T are chosen such 50

5 that there exists no T α T M resp. T β T M with a non-trivial idempotent element d ]a α, b α [ resp. d ]a β, b β [ left. As such we can still distinguish two different cases: Case. For all β J there exists some α, α I such that a α = a β and b α = b β or vice versa, expressing that the ordinal sums are based mainly on the same structure but one has at least one more summand not coinciding by structure with any of the summands of the other ordinal sum. Case. The elements a α, b α, α I, and a β, b β, β J, need not be related at all. As mentioned previously the domination property induces the order on t-norms, i.e., if T T then also T T. Moreover, since all t-norms are bounded from above by the minimum we can state the following lemma. Lemma 6. Consider two ordinal sum t-norms T = ( a α, b α, T α ) α I and T = ( a β, b β, T β ) β J. If T dominates T, necessarily T (x, y) = min(x, y) whenever T (x, y) = min(x, y). Although this lemma is trivial, it provides some insight into the structure of the involved ordinal sum t-norms. So if T dominates T it must necessarily consist of more regions where it acts as the minimum than T, meaning that necessarily in Case as introduced above it holds that I J. Focussing on the idempotent elements of ordinal sums, we note that for a continuous ordinal sum t- norm T = ( a α, b α, T α ) α A the set of idempotent elements ID T coincides with [0, ]\ α A ]a α, b α [. Which influence do these elements have on the property of domination? Proposition 7. Consider two ordinal sum t- norms, namely, T = ( a α, b α, T α ) α I and T = ( a β, b β, T β ) β J, and some element d [0, ]. If T dominates T then we get the following: (i) If d ID T, then also T (d, d) ID T. (ii) If d ID T, then also d ID T. What are the consequences of this proposition for the two different cases w.r.t. to the boundary elements of the families of the pairwise disjoint open 5 subintervals? Firstly, all boundary elements of T are idempotent elements of T, i.e., either boundary elements themselves or elements of some domain where T acts as T M. Secondly, for each idempotent element d of T we know that also d (n) T, n N, is an idempotent element of T, once again being either a boundary element itself or element of some domain where T acts as T M. Let us finally have a look at two families of t- norms being families of ordinal sum t-norms with only one summand but varying boundary elements. Example 8. The members of the family of Dubois-Prade t-norms, first introduced in [5], are defined by T DP (x, y) = ( 0,, T P ) and are ordinal sums with the product t-norm as a single summand. If = 0 the corresponding Dubois-Prade t-norm coincides with T M which dominates all t- norms. Therefore we assume that ]0, ]. Due to the induced order of the t-norms we necessarily have that if T DP T DP then. Is this condition also sufficient? If = the domination property is trivially fulfilled. Therefore, suppose that <. For better readability we will denote T DP (resp. T DP ) by T (resp. T ). For each T i, i {, }, the set of idempotent elements is given by ID Ti = {0} [ i, ] and necessarily for ID T also T (, ) ID T. But 0 T (, ) = T P (, ) = < due to the strict monotonicity of the involved T P. Therefore, T (, ) is not an element of ID T showing that there exists no domination property within the family in accordance with the involved parameter except for = 0, namely T M. Example 9. Similarly the family of Mayor- Torrens t-norms (T MT ) [0,] can be expressed as a family of ordinal sum t-norms ( 0,, T L ) [0,] [6]. Again as above T MT T MT implies. For our further investigations, we will denote by T (resp. T ) the involved T MT (resp. T MT ). The set of idempotent elements are of the following form ID Ti = {0} [ i, ]. We know that ID T and therefore necessarily also T (, ) ID T. Since

6 T (, ) T M (, ) = we can conclude that T (, ) = 0 or T (, ) = where the latter contradicts the fact that T L possesses only trivial idempotent elements. This leads us to T (, ) = 0 being equivalent to as a necessary condition for the domination property within this family. For all such t-norms choose some x fulfilling < x < + 4 and put u = v = y = x then T (T (x, y), T (u, v)) = 0 and T (T (x, u), T (y, v)) = x > 0 contradicting () for T T. Therefore, also in the Mayor-Torrens family of t-norms no domination appears except when T0 MT = T M is involved. Acknowledgments The support of this work by the EU COST Action 74 (TARSKI: Theory and Applications of Relational Structures as Knowledge Instruments) is gratefully acknowledged. References [] R.M. Tardiff. Topologies for probabilistic metric spaces, Pacific J. Math., vol. 65, pp. 33 5, 976. [] B. Schweizer and A. Sklar. Probabilistic Metric Spaces. North-Holland, New York, 983. [3] S. Saminger, R. Mesiar, and U. Bodenhofer. Domination of aggregation operators and preservation of transitivity, Internat. J. Uncertain. Fuzziness Knowledge-Based Systems, vol. 0/s, pp. 35, 00. [4] R. Mesiar and S. Saminger. Commuting aggregation operators, in Proceedings of EUSFLAT 003, Zittau (Germany), pp , 003. [5] S. Saminger. Aggregation in Evaluation of Computer-Assisted Assessment, volume C 44 of Schriftenreihe der Johannes-Kepler- Universität Linz. Universitätsverlag Rudolf Trauner, 005. [6] E.P. Klement, R. Mesiar, and E. Pap. Triangular Norms, volume 8 of Trends in Logic. Studia Logica Library. Kluwer Academic Publishers, Dordrecht, 000. [7] P.S. Mostert and A.L. Shields. On the structure of semigroups on a compact manifold with boundary, Ann. of Math., vol. 65, pp. 7 43, 957. [8] A. Pradera, E. Trillas, and E. Castiñeira. On the aggregation of some classes of fuzzy relations, in B. Bouchon-Meunier, J. Gutiérrez-Ríos, L. Magdalena, and R.R. Yager, editors, Technologies for Constructing Intelligent Systems : Tasks. Springer, 00. [9] W. Jarczyk and J. Matkowski. On mulholland s inequality, Proc. Amer. Math. Soc., vol. 30, pp , 00. [0] H.P. Mulholland. On generalizations of Minkowski s inequality in the form of a triangle inequality, Proc. London Math. Soc., vol. 5, pp , 950. [] R.M. Tardiff. On a functional inequality arising in the construction of the product of several metric spaces, Aequationes Math., vol. 0, pp. 5 58, 980. [] R.M. Tardiff. On a generalized Minkowski inequality and its relation to dominates for t-norms, Aequationes Math., vol. 7, pp , 984. [3] H. Sherwood. Characterizing dominates on a family of triangular norms, Aequationes Math., vol. 7, pp , 984. [4] P. Sarkoci. Domination in the families of Frank and Hamacher t-norms, Kybernetika, vol. 4, pp , 005. [5] D. Dubois and H. Prade. New results about properties and semantics of fuzzy settheoretic operators, in P.P. Wang and S.K. Chang, editors, Fuzzy Sets: Theory and Applications to Policy Analysis and Information Systems, pp Plenum Press, New York, 980. [6] G. Mayor and J. Torrens. On a family of t-norms, Fuzzy Sets and Systems, vol. 4, pp. 6 66, 99. 5