Contour Lines: The Lost World

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1 Contour Lines: The Lost World K. Maes Department of Applied Mathematics, Biometrics and Process Control, Ghent University, Coupure links 653, B-9000 Gent, Belgium B. De Baets Department of Applied Mathematics, Biometrics and Process Control, Ghent University, Coupure links 653, B-9000 Gent, Belgium Abstract Contour lines totally fix the structure of leftcontinuous t-norms. For each t-norm, these contour lines are determined by the corresponding residual implicator. Most properties involving residual implicators can now easily be translated into properties involving contour lines. As the portation law expresses associativity, we dispose of a powerful tool for constructing left-continuous t-norms. In particular, we focus on the decomposition and construction of t-norms that are rotation invariant w.r.t. one of their contour lines. Keywords: Left-continuous t-norm, rotationinvariant t-norm, portation law, contour line, associativity 1 Introduction After Schweizer and Sklar [12] introduced triangular norms in order to generalize the triangle inequality towards probabilistic metric spaces, t- norms have been widely used in fuzzy set theory. So far only the class of continuous t-norms has been characterized. However, diverse construction methods have been proposed for creating new t-norms. Most of these methods start from a known t-norm on which a bunch of operations such as rotations, annihilations and rescalings is performed. In other cases multiple t-norms are merged into a brand new t-norm. Unfortunately, these constructions were only elaborated for restricted classes of t-norms. In this contribution we try to analyse the basic structure of leftcontinuous t-norms. A left-continuous t-norm is completely fixed by its contour lines. Expressing its associativity by means of these contour lines, enables us to gain more insight in the structure of the t-norm in question. We are able to interpret the rotation and rotation-annihilation construction of Jenei [7, 8, 9, 10] into a new and more general framework, covering all known examples of rotation-invariant t-norms. Furthermore, working with contour lines allows us to tackle other construction methods. We briefly indicate our further results. Throughout this contribution, we use the standard notations for the prototypical t-norms and t-conorms [11]. We say that a t-norm T has zero divisors if there exists a couple (x, y) ]0, 1] 2 such that T (x, y) = 0. For any t-norm T and any [0, 1]- automorphism φ (i.e. increasing permutation of [0, 1]), the φ-transform of T, defined by T φ (x, y) = φ 1 (T (φ(x), φ(y))), is a t-norm as well. A negator N is a decreasing [0, 1] [0, 1] mapping that satisfies the boundary conditions 0 N = 1 and 1 N = 0. Remark that we use the exponential notation x N to denote N(x). A negator is called involutive if (x N ) N = x, for all x [0, 1]. The standard negator N is defined by x N = 1 x. 2 Rotation-invariant t-norms In several studies dealing with t-norms, it seems that the t-norms in question should be leftcontinuous. In logic for example, where the implication is defined as the residiuum of the conjunction, left-continuous t-norms ensure the definability of the t-norm-based residual implicator [1]. 395

2 The residual implicator 1 of a t-norm T is defined by I T : [0, 1] 2 [0, 1] (x, y) sup{t [0, 1] T (x, t) y}. T is left-continuous if and only if the following condition (Galois connection) [3] is satisfied for every (x, y, z) [0, 1] 3 : T (x, z) y I T (x, y) z. (1) A t-norm T is called rotation invariant w.r.t. an involutive negator N if for every (x, y, z) [0, 1] 3 it holds that T (x, y) z T (y, z N ) x N. This property was first described by Fodor [2]. Jenei emphasized its geometrical interpretation by referring to it as the rotation invariance of T [5]. Rotation-invariant t-norms can easily be spotted by means of the following theorem. Theorem 1 [5] Consider a left-continuous t- norm T and an involutive negator N. The following statements are equivalent: 1. T is rotation invariant w.r.t. N; 2. I T (x, y) = z T (x, y N ) = z N, for every (x, y, z) [0, 1] 3 ; 3. I T (x, 0) = x N, for every x [0, 1]. The second assertion in the theorem is also known as the self quasi-inverse property. The negator N : [0, 1] [0, 1] : x I T (x, 0) is called the residual negator of T. Rotation invariance itself also implies the left-continuity of the t-norm in question [5]. Despite their importance, until recently little was known concerning the structure of left-continuous t-norms. Jenei provided a real breakthrough by introducing his rotation construction, rotation annihilation construction and embedding method 1 Strictly speaking, the term residual is only appropriate for a left-continuous t-norm T. [7, 8, 9, 10]. Starting from known t-norms, fulfilling some rather weak conditions, Jenei obtained rotation-invariant left-continuous t-norms by rescaling, rotating and annihilating the original ones. The Jenei t-norm family (Tλ J) λ [0,1/2], Tλ J (x, y) = 0, if x + y 1, x + y 1 + λ, if x + y > 1, and (x, y) ]λ, 1 λ] 2, min(x, y), elsewhere, (2) consists of left-continuous rotation-invariant (w.r.t. N ) t-norms [7], obtained by applying the rotation annihilation construction on T M and T L. Note that T0 J = T L and T1/2 J = T nm. 3 Contour lines Starting from a left-continuous [0, 1] 2 [0, 1] mapping, we can easily redefine left-continuous t- norms by means of contour lines. The neutral element, monotonicity and commutativity are easily translated to conditions on contour lines. Only associativity is quite difficult to fathom. We will show that associativity is equivalent to the portation law, which can be expressed by means of contour lines. Furthermore, we will provide necessary and sufficient conditions such that a leftcontinuous t-norm is rotation invariant w.r.t. one of its contour lines. In the basic logic BL [4] the following axioms are used to express residuation: (φ (ψ χ)) ((φ & ψ) χ), (3) ((φ & ψ) χ) (φ (ψ χ)), (4) with φ, ψ and χ formulas of the logic. The evaluation of both axioms in a BL-algebra ([0, 1],,, T, I T, 0, 1), with T a left-continuous t- norm and I T its residual implicator, must be eqaul to 1: I T (I T (x, I T (y, z)), I T (T (x, y), z)) = 1, (5) I T (I T (T (x, y), z), I T (x, I T (y, z))) = 1, (6) for every (x, y, z) [0, 1] 3. Since I T (x, y) = 1 = I T (y, x) only occurs for x = y, we can derive from 396

3 Eqs. (5) and (6) that I T (x, I T (y, z)) = I T (T (x, y), z) (7) holds for every (x, y, z) [0, 1] 3. The latter is also referred to as the portation law [5]. For every left-continuous t-norm T, (7) is always fulfilled. For every left-continous t-norm T, its residual implicator I T geometrically determines its contour lines : [0, 1] [0, 1] : x I T (x, a), with a [0, 1]. It will be clear from the context which t-norm T we are considering. Note that (x) = 1 if and only if x a. The geometrical perception of the residual implicator enables us to rewrite the portation law, expressed by (7), as follows: (T (x, y)) = C Ca(x)(y), (8) for every (x, y, a) [0, 1] 3. The symmetry of T implies that C Ca(x)(y) = C Ca(y)(x), which is also known as the exchange principle. Moreover, it is easily seen that (8) is trivially true when T (x, y) a. The portation law, expressed by (8), plays a profound role when dealing with a left-continuous t-norm T as it is equivalent to its associativity. Theorem 2 An increasing and left-continuous mapping T : [0, 1] 2 [0, 1] that fulfills for every x [0, 1] the boundary condition T (x, 0) = 0 is associative if and only if T fulfills the portation law. Moreover, the portation law enables us to gain a clearer understanding of the structure of leftcontinuous t-norms. Property 1 Consider a left-continuous t-norm T and a [0, 1[. The following statements are equivalent: 1. ( (x)) = x, for every x a; 2. C b (x) = C Ca(x)( (b)), for every b a. Given a negator N and a [0, 1[, define the mapping N a : [0, 1] [a, 1] by 1, if x a, ( ) x Na = x a N a + (1 a), elsewhere. 1 a In particular N 0 = N and x N 1 = 1, for every x [0, 1]. The function N a can be understood as a trivial extension of a rescaled version of N. We call N a involutive if (x Na ) Na = x, for every x a. Obviously, N a will be involutive if and only if N is an involutive negator. A t-norm T is said to be rotation invariant w.r.t. an involutive N a if T (x, y) z T (y, z Na ) x Na, for every (x, y, z) [a, 1] 3. The latter expresses that some kind of linear rescaling of T [a,1] 2 to [0, 1] 2 is rotation-invariant w.r.t. N. Applying that every rotation-invariant (w.r.t. an involutive negator N) t-norm is necessarily left-continuous [5], immediately leads to the following theorem. Theorem 3 Consider an involutive negator N and a [0, 1[. If a t-norm T is rotation invariant w.r.t. N a, then it is left-continuous on D = {(x, y) [0, 1] 2 x Na < y}. We are now able to rewrite and extend Theorem 1 as follows: Theorem 4 Consider a left-continuous t-norm T, an involutive negator N and a [0, 1[. The following statements are equivalent: 1. T is rotation invariant w.r.t. N a ; 2. C x (y) = C y Na (x Na ), for every (x, y) [a, 1] 2 ; 3. C b (x) = y T (x, b Na ) = y Na, for every (x, y, b) [a, 1] 3 ; 4. x Na = (x), for every x [a, 1]. Obviously, every continuous, Archimedian t-norm (i.e. a t-norm of the form (T P ) φ or (T L ) φ, with φ a [0, 1]-automorphism) is rotation invariant w.r.t. each of its countour lines. Property 2 Consider a left-continuous t-norm T and a [0, 1[. is involutive if and only if T (x, y) = (C Ca(x)(y)), (9) for every (x, y) fulfilling y > (x). 397

4 4 Constructing rotation-invariant t-norms If a t-norm T is rotation invariant w.r.t. an involutive decreasing [0, 1] [a, 1] mapping N a, then N a = and T is left-continuous strictly above its contour line. Assuming some continuity conditions, we will now attempt to reconstruct T in the area D strictly above. 1 a II IV I III 0 0 a 1 C (x) C (x) a Figure 1: The partition D = D I D II D III D IV. First, partition D into four parts as pictured in Figure 1: D I = {(x, y) ]β, 1] 2 y > C β (x)}, D II = {(x, y) ]a, β] ]β, 1] y > (x)}, D III = {(x, y) ]β, 1] ]a, β] y > (x)}, D IV = {(x, y) ]β, 1] 2 y C β (x)}. Denote β the unique fixpoint of (i.e. (β) = β). As will become clear, area D I is crucial in the construction and decomposition of rotationinvariant t-norms. Theorem 5 Consider a t-norm T that is rotation invariant w.r.t. one of its contour lines. Let σ be an arbitrary increasing bijection from [β, 1] to [0, 1], with β the unique fixpoint of. Then the [0, 1] 2 [0, 1] mapping T, defined by T (x, y) = σ(max(β, T (σ 1 [x], σ 1 [y]))), is a left-continuous t-norm. Denoting the contour lines of T by C b it also holds that T (x, y) = [ ] σ 1 T (σ(x), σ(y)), if (x, y) D I, (σ 1 Cσ(Ca(x))(σ(y)), if (x, y) D II, (σ 1 Cσ(Ca(y))(σ(x)), if (x, y) D III. (10) Interpreting this theorem, it follows that D I is in fact the rescaling of the area of T strictly above. Once T DI is known, it can be used to create T DII, which in turn can be used to construct T DIII. This leaves us to figure out how to fill up area D IV. In case D IV is empty we know how T D is constructed by means of a left-continuous t-norm T (i.e. the rescaling of T DI ) without zero divisors. Conversely, we wonder when an arbitrary left-continuous t-norm T without zero divisors ensures that T D fulfills the properties of a t-norm. Theorem 6 Consider an involutive negator N and a [0, 1[. Let T be a left-continuous t-norm without zero divisors and with contour lines C b. Define := N a and C β := ( ) β, with β the unique fixpoint of. Then T, defined by (10), is a symmetrical, increasing, associative mapping fulfilling T (x, 1) = x, for every x ]a, 1]. In particular, for a = 0 this theorem implies that T itself is a t-norm. Corollary 1 Consider an involutive negator N. Let T be a left-continuous t-norm without zero divisors and with contour lines C b. Define := N and C β := ( ) β, with β the unique fixpoint of. Then the [0, 1] 2 [0, 1] mapping T, defined by T (x, y) = 0, if (x, y) D, [ ] σ 1 T (σ(x), σ(y)), if (x, y) D I, (σ 1 Cσ(C0 (x))(σ(y)), if (x, y) D II, (σ 1 Cσ(C0 (y))(σ(x)), if (x, y) D III, is a rotation-invariant (w.r.t. N) t-norm. Our approach in the previous corollary is identical to the rotation construction of Jenei [7, 9]. On 398

5 the other hand, if T has zero divisors and we still want that T DI D IV is just a rescaling of T, then D IV must be a square. This case has also been covered by Jenei s rotation construction [7, 9]. For t-norms T that are continuous on D we will show that T D is totally fixed by T DI. Theorem 7 Consider a t-norm T that is rotation invariant w.r.t. one of its contour lines and that is continuous on D. Let σ be an arbitrary increasing bijection from [β, 1] to [0, 1], with β the unique fixpoint of. Then the [0, 1] 2 [0, 1] mapping T, defined by T (x, y) = σ(max(β, T (σ 1 [x], σ 1 [y]))), is a continuous t-norm. Denoting the contour lines of T by C b it also holds that T (x, y) = [ ] σ 1 T (σ(x), σ(y)), if (x, y) D I, (σ 1 Cσ(Ca(x))(σ(y)), if (x, y) D II, (σ 1 Cσ(Ca(y))(σ(x)), if (x, y) D III, (σ 1 T (σ(cβ (x)), σ(c β (y))), if (x, y) D IV. (11) Conversely, every continuous t-norm T can be used to construct T D. Theorem 8 Consider an involutive negator N and a [0, 1[. Let T be a continuous t-norm with contour lines C b. Define := N a and C β := ( ) β, with β the unique fixpoint of. Then T, defined by (11), is a symmetrical, increasing, associative mapping fulfilling T (x, 1) = x, for every x ]a, 1]. In case a = 0 we can create all rotation invariant (w.r.t. an involutive negator N) t-norms T that are continuous strictly above = N. Corollary 2 Consider an involutive negator N. Let T be a continuous t-norm with contour lines C b. Define := N and C β := ( ) β, with β the unique fixpoint of. Then the [0, 1] 2 [0, 1] mapping T, defined by T (x, y) = 0, if (x, y) D, [ ] σ 1 T (σ(x), σ(y)), if (x, y) D I, (σ 1 Cσ(C0 (x))(σ(y)), if (x, y) D II, (σ 1 Cσ(C0 (y))(σ(x)), if (x, y) D III, (σ 1 T (σ(cβ (x)), σ(c β (y))), if (x, y) D IV, is a rotation-invariant (w.r.t. N) t-norm. Since (T L ) φ, with φ a [0, 1]-automorphism, is continuous and rotation invariant w.r.t. each of its contour lines, Theorem 7 can be used to decompose such a t-norm. Take a [0, 1[ and let β be the unique fixpoint of. Then (T L ) φ D\DI is uniquely determined by (T L ) φ DI. In particular, (T L ) φ DI must be the rescaling of the strictly positive part of a t-norm (T L ) φ, with φ some [0, 1]- automorphism. Analogous results hold for (T P ) φ. We only need to restrict a to ]0, 1[. 5 Further research In the previous section we only presented a selection of our results. We are able to interpret Jenei s rotation-annihilation construction [8, 9] into our framework and we can uniquely determine T DIV under other specific extra conditions. Examining numerous examples, we know that T DIV is not always uniquely fixed by means of T DI. Also, not every left-continuous t-norm T is appropriate for constructing T D. Besides all these construction and decomposition results, we also used contour lines to derive necessary and sufficient conditions for the annihilation of t-norms. Our findings largely extend the existing knowledge (see e.g. [6]). These results will be reported upon in the near future. Given the numerous new insights contour lines provided on the structure of t-norms, we intend to investigate their importance for other aggregation operators. Furthermore, we plan to have a deeper look on the close relationship between contour lines and residual implicators, wondering 399

6 which new properties concerning contour lines are of interest for residual implicators. A more profound study of contour lines can be expected in future contributions. References [1] F. Esteva and L. Godo, Monoidal t- norm based logic: towards a logic for leftcontinuous t-norms, Fuzzy Sets and Systems, vol. 124, pp , [11] E. Klement, R. Mesiar and E. Pap, Triangular Norms, ser. Trends in Logic, vol. 8. Kluwer Academic Publishers, [12] B. Schweizer and A. Sklar, Probabilistic Metric Spaces. New York: Elsevier Science, [2] J. Fodor, A new look at fuzzy connectives, Fuzzy Sets and Systems, vol. 57, pp , [3] J. Fodor and M. Roubens, Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer Academic Publishers, [4] P. Hájek, Metamathematics of Fuzzy Logic, ser. Trends in Logic, vol. 4. Kluwer Academic Publishers, [5] S. Jenei, Geometry of left-continuous t- norms with strong induced negations, Belg. J. Oper. Res. Statist. Comput. Sci., vol. 38, pp. 5 16, [6], New family of triangular norms via contrapositive symmetrization of residual implications, Fuzzy Sets and Systems, vol. 110, pp , [7], Structure of left-continuous triangular norms with strong induced negations. (I) rotation construction. J. Appl. Non-Classical Logics, vol. 10, pp , [8], Structure of left-continuous triangular norms with strong induced negations. (II) rotation-annihilation construction. J. Appl. Non-Classical Logics, vol. 11, pp , [9], A characterization theorem on the rotation construction for triangular norms, Fuzzy Sets and Systems, vol. 136, pp , [10], How to construct left-continuous triangular norms - state of the art, Fuzzy Sets and Systems, vol. 143, pp ,