Triple Rotation: Gymnastics for T-norms

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1 Triple Rotation: Gymnastics for T-norms K.C. Maes Department of Applied Mathematics, Biometrics and Process Control, Ghent University, Coupure links 653, B-9 Gent, Belgium B. De Baets Department of Applied Mathematics, Biometrics and Process Control, Ghent University, Coupure links 653, B-9 Gent, Belgium Abstract Given an involutive negator N and a leftcontinuous t-norm T whose contour line C is continuous on ],], we build a rotationinvariant t-norm from a rescaled version of T and its left, right and front rotation. Depending on the involutive negator N and the set of zero divisors of T, some reshaping of the rescaled version of T may occur during the rotation process. The rescaled version of T itself can be understood as the β-zoom of the newly constructed rotation-invariant t-norm, with β the unique fixpoint of N. Keywords: Triple Rotation Method, Rotation-invariant T-norm, Contour Line, Companion, Zoom. Introduction Studying the structure of an increasing [,] 2 [,] function T, the use of contour lines and zooms has proven to be very fruitful (see e.g. [8, 3, 4, 5, 6]). Also, the companion is indispensable to describe for example rotation-invariant t-norms (see e.g. [4, 6]). Note that we use the standard notations for the prototypical t-norms and t-conorms [9, ]. Contour lines Contour lines of an increasing [,] 2 [,] function T are defined as the lower, upper, left or right limits of its horizontal cuts, i.e. the intersections of its graph by planes parallel to the domain [,] 2. Although there exist four different types of contour lines, those determined by the upper limits of the horizontal cuts are of particular interest for the study of rotationinvariant t-norms [4]. Definition [3] Let a [, ]. The contour line C a of an increasing [,] 2 [,] function T is the [, ] [, ] function defined by C a (x) = sup{t [,] T(x,t) a}. () For a left-continuous t-norm T, the contour line C a equals the partial function I T (,a) of the residual implicator I T (see e.g. [4]). In particular, the zero contour line C then coincides with the residual negator N T = I T (,). Contour lines of a continuous t-norm T are also called level functions []. The companion A second useful tool to study an increasing [,] 2 [,] function T is its companion Q. Definition 2 [4] The companion Q of an increasing [,] 2 [,] function T is the [,] 2 [,] function defined by Q(x,y) = sup{t [,] C t (x) y}. We have shown in [4] that Q(x,y) = inf{t(x,u) u ]y,]} (with inf = ). This property allows to construct the graph of Q from the graph of T. Clearly, Q(x,y) = T(x,y) whenever T(x, ) is right continuous in y [, [. Every left-continuous, increasing [,] 2 [,] function T that has absorbing element is determined by its companion Q. Zooms Finally, every increasing [,] 2 [,] function T is trivially described by its associated set of zooms. Definition 3 [6] Let T be an increasing [,] 2 [,] function and take (a,b) [,] 2 such that a < b and T(b,b) b. Consider an [a,b] [,] isomorphism σ. The (a,b)-zoom T (a,b) of T is the [,] 2 [,] function defined by T (a,b) (x,y) = σ [ max ( a,t(σ [x],σ [y]) )]. If b = we simply talk about the a-zoom T a of T.

2 The graph of T (a,b) is determined by the rescaling of the set {(x,y,t(x,y)) (x,y) [a,b] 2 a < T(x,y)} (zoom in) into the unit cube (zoom out). Whenever T(b,b) a, the function T (a,b) is trivially constant: T (a,b) (x,y) = a, for every (x,y) [,] 2. For b = the boundary condition T(,) is always fulfilled such that the a-zoom of T is defined for every a <. Zooms are extremely suited to study an increasing function T that satisfies T T M. The restriction T(b,b) b (Definition 3) then trivially holds. Recall that a t-subnorm T is a [,] 2 [,] function satisfying all t-norm properties but the neutral element. Instead T T M must hold [7]. Theorem [6] Consider (a,b) [,] 2 such that a < b. Then the (a,b)-zoom of a t-subnorm is a t- subnorm and the a-zoom of a t-norm is a t-norm. 2 The triple rotation method It is well known that left-continuous t-norms ensure the definability of the t-norm-based residual implicator. Therefore they are of great interest to people working on monoidal t-norm based logic (MTL logic) [2] and involutive monoidal t-norm based logic (IMTL logic) [, 2]. The latter requires the involutivity of the residual negator N T = C. We have shown [3, 4] that the involutivity of C is equivalent with its continuity and with the rotation invariance of the left-continuous t-norm T considered. Definition 4 [5] Let N be an involutive negator. An increasing [,] 2 [,] function T is called rotation invariant w.r.t. an involutive negator N (i.e. an involutive decreasing [,] [,] function) if for every (x,y,z) [,] 3 it holds that T(x,y) z T(y,z N ) x N. (2) Jenei has proven that every t-norm T that is rotation invariant w.r.t. an involutive negator N is necessarily left continuous and N T = N [5]. Therefore, we briefly call a t-norm rotation invariant if it is left continuous and has a continuous contour line C. Note that the continuity of C does not necessarily imply the left continuity of T [4]. Based on contour lines, the companion and zooms, we have presented in [6] a natural method for decomposing a rotation-invariant t-norm T. In case the contour line C β of T, with β the unique fixpoint of C, is continuous on ]β,], there exists a unique decomposition. In this contribution we transform our decomposition method into a straightforward construction tool for rotation-invariant t-norms. The presented results extend our work from [5] and comprise to a large extent the rotation and rotation-annihilation construction of Jenei [7]. We assume the following setting: T: an arbitrary left-continuous t-norm (with contour lines C a and companion Q) such that C is continuous on ],] and Q is commutative on [,α[ 2, with α = inf{t [,] C (t) = }. N: an arbitrary involutive negator with fixpoint β. σ: an arbitrary [β, ] [, ] isomorphism. M: the decreasing [,] [,β] function defined by x M = whenever x [,β[ and x M = σ [C (σ[x])] whenever x [β,]. D: the area {(x,y) [,] 2 x N < y} = D I D II D III D IV, with D I = {(x,y) ]β,] 2 x M < y}, D II = {(x,y) ],β] ]β,] x N < y}, D III = {(x,y) ]β,] ],β] x N < y}, D IV = {(x,y) ]β,[ 2 y x M }. Note that the choice of T, N and σ fixes M and D. Theorem 2 The [,] 2 [,] function R3(T,N) defined by R3(T, N)(x, y) = σ [T(σ[x],σ[y])], if (x,y) D I, ( [ σ C σ[xn ](σ[y]) ]) N, if (x,y) DII, ( [ σ C σ[yn ](σ[x]) ]) N, if (x,y) DIII, ( [ σ Q ( C (σ[x]),c (σ[y]) )]) N, if (x,y) DIV,, if (x,y) D, (3) is a rotation-invariant t-norm. R3(T,N) is the only left-continuous t-norm that has N as a contour line (a = ) and has β-zoom R3(T,N) β = T. In [5] we showed that R3(T,N) DII and R3(T,N) DIII are determined by the (transformed) left and right rotation of R3(T,N) DI around the axis through the points (,, ) and (,, ). As will become clear from the examples, R3(T,N) DIV is determined by the (transformed) front rotation of R3(T,N) DI ]β,σ (α)] 2 around the axis through the points (β,σ [α],β) and (σ [α],β,β). Note also that R3(T,N) DI is a rescaled version of the non-zero part of T. Inspired by these geometrical observations, we briefly call R3(T,N) the triple rotation of T based on N. The construction method itself is referred to as the triple rotation method. For the following examples we use the linear rescaling function ς : x (x β)/( β). Any other rescaling function entails a transformation of the procured t- norm.

3 ... (b) R3(TM, N ) = T nm (a) TM. (c) Contour plot of R3(TM, N )... (e) R3(TP, N ) (d) TP. (f) Contour plot of R3(TP, N ) Figure : The triple rotation of TM and TP based on N. A first class of examples is obtained by considering those t-norms T that have no zero-divisors (i.e. < T (x, y), for every (x, y) ], ]2 ). In this case α =, σ [α] = β and DIV =. The triple rotation method then coincides with the rotation construction of Jenei [6]. In Fig., for example, we apply the triple rotation method to the minimum operator TM and the algebraic product TP. The triple rotation R3(TM, N ) of TM based on the standard negator N equals the nilpotent minimum T nm. The bold black lines in Figs. (a) and (d) indicate the corresponding zero contour lines. The bold black lines in all other subfigures visualize the partition D = DI DII DIII DIV. Secondly, the triple rotation method can be performed on most of the rotation-invariant t-norms. In this case α = = σ [α] [3] and we can rewrite Eq. (3) in a more feasible form [4]: R3(T, N )(x, y) = σ [T (σ[x], σ[y])], N σ C T C (σ[xn ]), σ[y], N, σ C T σ[x], C (σ[y N ]) N, σ Q C (σ[x]), C (σ[y]), if (x, y) DI, if (x, y) DII, if (x, y) DIII, if (x, y) DIV, if (x, y) 6 D. However, for the triple rotation method to yield a t-norm it is absolutely necessary that the companion Q of T is commutative on [, [2 [5]. The rotation-invariant t-norms depicted in Figs. 2(a), 2(d) and 2(g) satisfy this mandatory condition. They are the φ-transforms of the triple rotations R3(TM, N ), R3(TP, N ) and R3(TL, N ), with φ the automorphism defined by φ(x) := x3/5. Recall that the φ-transform Tφ of a t-norm T is the t-norm defined by Tφ (x, y) = φ [T (φ[x], φ[y])]. Let N be the involutive negator defined by q if x [, 3 ] 3 9 x, q (4) xn := (x 3 )2, if x [ 3, 32 ] q (x 2 )2, if x [ 32, ]. 9 3 In Figs. 2(b), 2(e) and 2(h) we apply the triple rotation method based on N to the φ-transforms R3(TM, N )φ, R3(TP, N )φ and R3(TL, N )φ. The obtained rotationinvariant t-norms cannot be described by the rotation construction nor by the rotation-annihilation construction of Jenei [5]. As can be seen in Figs. 2(b), 2(e) and 2(h), if N 6= N or C 6= N, the left, right and front rotation of R3(T, N ) DI have to be reshaped to

4 . (a) R3(TM, N )φ (b) R3(R3(TM, N )φ, N ). (c) Contour plot of R3(R3(TM, N )φ, N ). (d) R3(TP, N )φ (e) R3(R3(TP, N )φ, N ). (f) Contour plot of R3(R3(TP, N )φ, N ). (g) R3(TL, N )φ (h) R3(R3(TL, N )φ, N ). (i) Contour plot of R3(R3(TL, N )φ, N ) Figure 2: The triple rotation of R3(TM, N )φ, R3(TP, N )φ and R3(TL, N )φ based on N. fit into the areas DII, DIII and DIV, respectively. The involutive negator N and the contour line C of T are responsible for this reshaping. Note also that in general R3(T, N )(β, ) = N M = R3(T, N )(, β). Therefore, the t-norms R3(T, N ) visualized in Figs. 2(b), 2(e) and 2(h) have identical partial functions R3(T, N )(, β) = R3(T, N )(β, ), with β = the fixpoint of N. Indeed, their associated functions M (C = Nφ ) and N = N are identical. Finally, if α ], [, then T is necessarily an ordinal sum of a rotation-invariant t-norm whose companion is commutative on [, [2 and an arbitrary leftcontinuous t-norm. The latter largely follows from the following characterization. Theorem 3 [6] Consider a left-continuous t-norm T and take a [, ] such that a < θ := inf{t [, ] Ca (t) = a}. Then the following assertions are equivalent:

5 . (a) T (b) R3(T, N ). (c) Contour plot of R3(T, N ). (d) T2 (e) R3(T2, N ). (f) Contour plot of R3(T2, N ). (g) T3 (h) R3(T3, N ). (i) Contour plot of R3(T3, N ) Figure 3: The triple rotation of T, T2 and T3 based on N Ca is continuous on ]a, ]. Ca is involutive on ]a, θ[. Ca (]a, α[) =]a, θ[. T (a,θ) is a rotation-invariant t-norm. In Fig. 3 we present the triple rotation of the ordinal sums T :=, 2, R3(TM, N )φ, 2,, TM T2 :=, 2, R3(TP, N )φ, 2,, TP T3 :=, 2, R3(TL, N )φ, 2,, TL, based on the involutive negator N. Note that here α = 2. For the t-norms R3(T, N ) visual- ized in Figs. 3(b), 3(e) and 3(h) it clearly holds that R3(T, N ) DIV can be understood as a reshaped front rotation of R3(T, N ) DI ]β,ς ( 2 )]2, with β the fixpoint of N. The dashed lines in the figures indicate the area DI ]β, ς ( 2 )]2. The zooms N (R3(T, N ))((ς [ 2 ]),ς [ 2 ]) of these three t-norms R3(T, N ) (with T {T, T2, T3 }) are rotationinvariant t-norms, obtained by performing the triple rotation method on the rotation-invariant t-norms (R3(T, N ))(β,ς [ 2 ]) = T (, 2 ). For this latter construction the involutive negator ς N ς is used, with ς the linear rescaling function from [(ς ( 2 ))N, ς ( 2 ))] to [, ].

6 .. (a) R3(R3(T M, N) φ, N ) a (b) R3(R3(T P, N) φ, N ) a (c) R3(R3(T L, N) φ, N ) a Figure 4: The a-zooms of R3(R3(T M, N) φ,n ), R3(R3(T P, N) φ,n ) and R3(R3(T L, N) φ,n ), with a = (ς [φ ( 2 )])N. Besides the triple rotation method we can also use zooms to build new t-norms. Let a be the height of the two lowest jumps in Fig. 2(e). In Fig. 4 we visualize the a-zooms of the t-norms depicted in Figs. 2(b), 2(e) and 2(h). The bold black lines indicate the corresponding zero contour lines. References [] F. Esteva, J. Gispert, L. Godo, and F. Montagna, On the standard and rational completeness of some axiomatic extensions of the monoidal t- norm logic, Studia Logica 7 (22), [2] F. Esteva and L. Godo, Monoidal t-norm based fuzzy logic: towards a logic for left-continuous t- norms, Fuzzy Sets and Systems 24 (2), [3] J. Fodor, A new look at fuzzy connectives, Fuzzy Sets and Systems 57 (993), [4] J. Fodor and M. Roubens, Fuzzy Preference Modelling and Multicriteria Decision Support, Kluwer Academic Publishers, 994. [5] S. Jenei, Geometry of left-continuous t-norms with strong induced negations, Belg. J. Oper. Res. Statist. Comput. Sci. 38 (998), 5 6. [6] S. Jenei, A characterization theorem on the rotation construction for triangular norms, Fuzzy Sets and Systems 36 (23), [9] E.P. Klement, R. Mesiar, and E. Pap, Triangular Norms, Trends in Logic, Vol. 8, Kluwer Academic Publishers, 2. [] E.P. Klement, R. Mesiar, and E. Pap, Triangular norms. Position paper I: basic analytical and algebraic properties, Fuzzy Sets and Systems 43 (24), [] E.P. Klement, R. Mesiar, and E. Pap, Different types of continuity of triangular norms revisited, New Mathematics and Natural Computation (25), [2] L. Lianzhen and L. Kaitai, Involutive monoidal t- norm based logic and R logic, Internat. J. Intell. Systems 9 (24), [3] K.C. Maes and B. De Baets, A contour view on uninorm properties, Kybernetika 42 (3) (26), [4] K.C. Maes and B. De Baets, On the structure of left-continuous t-norms that have a continuous contour line, Fuzzy Sets and Systems 58 (27), [5] K.C. Maes and B. De Baets, The triple rotation method for constructing rotation-invariant t- norms, Fuzzy Sets and Systems, in press. [6] K.C. Maes and B. De Baets, Advances in the geometrical study of rotation-invariant t-norms, Lecture Notes in Computer Science (27), in press. [7] S. Jenei, How to construct left-continuous triangular norms - state of the art, Fuzzy Sets and Systems 43 (24), [8] S. Jenei, On the geometry of associativity, Semigroup Forum, to appear.

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