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On properties of uninorms with underlying t-norm and t-conorm given as ordinal sums Paweł Dryga´s∗
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Institute of Mathematics, University of Rzeszów, ul. Rejtana 16a, 35-959 Rzeszów, Poland
Abstract
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Uninorms as binary operations on the unit interval have been widely applied in the fuzzy set theory. This paper presents some properties of uninorm-like operations for which the underlying operations are given by ordinal sums. If the underlying operations of a uninorm are given by ordinal sums, then the Cartesian product of the union of two arbitrary intervals (one in [0, e] and the other in [e, 1], where e is the neutral element of the uninorm) is a set closed under the uninorm. When transposing such a Cartesian product to the unit square, one obtains a uninorm-like operation. As a result, we have described the uninorm-like operations for which the underlying operations are basic pseudo-t-norms and pseudo-t-conorms and one of them is idempotent. © 2009 Published by Elsevier B.V.
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MSC: 03E72; 03B52
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Keywords: Aggregation operations; Ordinal sum; Uninorm; Associative operations
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Binary operations on the unit interval have many applications in the fuzzy set theory. Specifically, they are applied as Q1 multivalued logical connectives (cf. [13,10]). To the contrary, they give examples of aggregation operations in the unit interval (cf. [20]). For this reason it is crucial to examine and characterize such operations. Uninorms are examples of binary operations on the unit interval. They were introduced by Yager and Rybalov [20] as a generalization of triangular norms and conorms letting the neutral element e to be an arbitrary point of the unit interval. Fodor et al. [9] examined a general structure of uninorms. The frame structure of uninorms and the characterization of representable uninorms were also presented in that paper. In case of e = 1 one obtains triangular norms and in case of e = 0 one obtains triangular conorms (see [13]). When e ∈ (0, 1) one obtains operations considered in many scientific articles (see [1–9,14,18–20]). Moreover, in the papers [8,11] the characterizations of continuous uninorms are given and in the papers [1,18] the descriptions of idempotent uninorms are presented. In this paper we associate the results from the papers [2,6] and we also consider more general class of operations than uninorms, i.e., we omit the assumption of the commutativity. The main aim of this paper is to present some properties of operations for which the underlying operations are given by ordinal sums. If the underlying operations of a uninorm are given by ordinal sums, then the Cartesian product of
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1. Introduction
∗ Tel.: +48 178721160.
E-mail address: [email protected]. 0165-0114/$ - see front matter © 2009 Published by Elsevier B.V. doi:10.1016/j.fss.2009.09.017 Please cite this article as: P. Dryga´s, On properties of uninorms with underlying t-norm and t-conorm given as ordinal sums, Fuzzy Sets and Systems (2009), doi: 10.1016/j.fss.2009.09.017
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the union of two arbitrary intervals, each from domain of different ordinal sum, is a set closed under the uninorm. Consequently, we describe such uninorm-like operations where the underlying operations are basic pseudo-t-norms and pseudo-t-conorms and one of them is idempotent. In Section 2 we present some notions and descriptions of the structure of uninorms and the construction of ordinal sum of semigroups. In Section 3 we characterize the uninorms which are continuous in the open unit square. Next, in Section 4, we present the structure of idempotent uninorms given in [18]. Moreover, we deal with operations which are locally internal on some subset of their domain and we give properties of the corresponding function and its influence on the structure of underlying operations (a t-norm and a t-conorm). Finally, in Section 5, we give some new properties of uninorms. 2. Notion of uninorms We will discuss the structure of binary operations in the class
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U(e) = {U : [0, 1]2 → [0, 1] : U is increasing, associative, with the neutral element e ∈ [0, 1]}. Definition 2.1 (see Yager and Rybalov [20], Sander [19]). Let e ∈ [0, 1].
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• An operation U ∈ U(e) is called a uninorm, if it is commutative. • A uninorm T (S) is called a t-norm (a t-conorm, respectively), if it has the neutral element e = 1 (e = 0, respectively). • An operation T ∈ U(1) (S ∈ U(0), respectively) is called a pseudo-t-norm (a pseudo-t-conorm, respectively).
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Lemma 2.3 (cf. Jenei [12], Drewniak and Dryga´s [5]). Let {[ak , bk ]}k∈T be a countable family of nonoverlapping, closed and proper subintervals of [0, 1]. Let T be an operation on [0, 1] defined by ⎧ x − ak y − ak ⎨ ak + (bk − ak )Tk if x, y ∈ (ak , bk ], , T (x, y) = (1) bk − ak bk − ak ⎩ min(x, y) otherwise,
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The operation given by (1) is called the ordinal sum of {([ai , bi ], Ti )}i∈T . Similarly, we may construct the operation with the neutral element 0. More information about ordinal sums one may find in [5] or [7]. Now, we present the frame structure of operations from the class U(e).
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where Tk : [0, 1]2 → [0, 1] are associative, increasing binary operations, such that Tk ≤ min for every k ∈ T . Moreover, we assume that operations Tk have neutral element e = 1 if bk = al and Tl is with zero divisor, or bk = 1. Then T is an operation from the class U(1). Moreover, the operation T is commutative if and only if for all k ∈ T the operations Tk are commutative.
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We may use these operations to construct new t-norms or t-conorms.
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Example 2.2 (cf. Klement et al. [13]). In Table 1 we present well-known triangular norms and conorms.
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Theorem 2.4 (cf. Fodor et al. [9]). If an operation U is in the class U(e), for e ∈ (0, 1), then there exist an operation T ∈ U(1) and an operation S ∈ U(0) such that ∗ T i n [0, e]2 , U= (2) S ∗ i n [e, 1]2 , where
x , x, y ∈ [0, e], e (x − e) , x, y ∈ [e, 1]. S ∗ (x, y) = −1 (S((x), (y))), (x) = (1 − e)
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T ∗ (x, y) = −1 (T ((x), (y))), (x) =
Let us denote the operations T and S mentioned in the above theorem by TU and SU and additionally 33
A(e) = [0, e) × (e, 1] ∪ (e, 1] × [0, e). Please cite this article as: P. Dryga´s, On properties of uninorms with underlying t-norm and t-conorm given as ordinal sums, Fuzzy Sets and Systems (2009), doi: 10.1016/j.fss.2009.09.017
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Table 1 Examples of basic t-norms and t-conorms t-Norm
t-Conorm
Name of the operations
TM (x, y) = min(x, y) TP (x, y) = xy TL (x, y) = max(x + y − 1, 0) ⎧ ⎨ x if y = 1 TD (x, y) = y if x = 1 ⎩ 0 otherwise
SM (x, y) = max(x, y) SP (x, y) = x + y − xy SL (x, y) = min(x + y, 1) ⎧ ⎨ x if y = 0 SD (x, y) = y if x = 0 ⎩ 1 otherwise
Lattice operations Algebraic operations Łukasiewicz operations Drastic operations
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Fig. 1. The structure of a uninorm.
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Lemma 2.5 (cf. Fodor et al. [9]). If U : [0, 1]2 → [0, 1] is an increasing operation with the neutral element e ∈ (0, 1), then
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Furthermore, if U is associative, then U (0, 1), U (1, 0) ∈ {0, 1}. The frame structure of uninorms, by Theorem 2.4 and Lemma 2.5, may be depicted in Fig. 1.
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Theorem 2.6 (cf. Li and Shi [14]). Let e ∈ (0, 1). If T is an arbitrary t-norm and S is an arbitrary t-conorm, then formula (2) with U = min or U = max in A(e) gives a uninorm.
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Example 2.7 (cf. Fodor et al. [9]). The formula 0 if x = 0 or y = 0, xy U (x, y) = if x > 0 and y > 0, (1 − x)(1 − y) + x y
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x, y ∈ [0, 1]
presents a uninorm with e = 21 , T (x, y) = x y/(2 − (x + y − x y)) and S(x, y) = (x + y)/(1 + x y), for x, y ∈ [0, 1]. Observe that T and S are arbitrary in Theorem 2.6, but here T and S are dual operations (cf. [13, p. 223]).
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3. Continuous operations in the open unit square 13
In the next theorem and Fig. 2 there is given the characterization of uninorm-like operations which are continuous on the open unit square. Please cite this article as: P. Dryga´s, On properties of uninorms with underlying t-norm and t-conorm given as ordinal sums, Fuzzy Sets and Systems (2009), doi: 10.1016/j.fss.2009.09.017
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Fig. 2. The operation U ∈ U (e) which is continuous in the open unit square with s > 0 (on the left) and t < 1 (on the right).
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Fig. 3. The structure of an idempotent operation belonging to the class U ∈ U (e).
Theorem 3.1 (Dryga´s [8]). Let e ∈ (0, 1) and U ∈ U(e) be continuous in (0, 1)2 . Then one of the following two cases holds:
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(i) There exist idempotent elements s ∈ [0, e) and p ∈ [0, s] of the operation U such that U |[0,1)2 is an ordinal sum of a continuous semigroup U |[0,s]2 with the neutral element s and a continuous group U |(s,1)2 with the Archimedean property. (ii) There exist idempotent elements t ∈ (e, 1] and q ∈ [t, 1] of the operation U such that U |(0,1]2 is a dual ordinal sum of a continuous semigroup U |[t,1]2 with the neutral element t and a continuous group U |(0,t )2 with the Archimedean property.
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Theorem 4.1 (Martín et al. [18]). Let e ∈ [0, 1]. An operation U ∈ U(e) is idempotent if and only if there exists a decreasing function g : [0, 1] → [0, 1] with g(e) = e, g(x) = 0 for all x > g(0), g(x) = 1 for all x < g(1) satisfying
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inf{y : g(y) = g(x)} ≤ g 2 (x) ≤ sup{y : g(y) = g(x)}, x ∈ [0, 1]
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and such that (see also Fig. 3) ⎧ ⎨ min(x, y) if y < g(x) or (y = g(x) and x < g 2 (x)), U (x, y) = max(x, y) if y > g(x) or (y = g(x) and x > g 2 (x)), ⎩ x or y if y = g(x) and x = g 2 (x). Please cite this article as: P. Dryga´s, On properties of uninorms with underlying t-norm and t-conorm given as ordinal sums, Fuzzy Sets and Systems (2009), doi: 10.1016/j.fss.2009.09.017
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Fig. 4. The operations U1 (on the left) and U2 (on the right) with the corresponding functions.
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Theorem 4.3 (Dryga´s [6]). Let e ∈ (0, 1). If an operation U ∈ U(e) is locally internal on A(e), i.e., U (x, y) ∈ {x, y} for x, y ∈ A(e), then (cf. Fig. 5): • there exists a decreasing function g : [0, 1] → [0, 1] with g(e) = e, g(x) = 0 for all x > g(0), g(x) = 1 for all x < g(1), satisfying inf{y : g(y) = g(x)} ≤ g 2 (x) ≤ sup{y : g(y) = g(x)} and such that
⎧ ⎨ min(x, y) if y < g(x) or (y = g(x) and x < g 2 (x)), U (x, y) = max(x, y) if y > g(x) or (y = g(x) and x > g 2 (x)), ⎩ x or y if y = g(x) and x = g 2 (x),
(x, y) ∈ A(e).
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Example 4.2. Let e ∈ (0, 1). For the uninorms U1 and U2 presented in Fig. 4 the corresponding functions are of the following form: 1 if x ∈ [0, e), e if x ∈ [0, e], gU1 (x) = gU2 (x) = e if x ∈ [e, 1], 0 if x ∈ (e, 1].
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• U |[0,e]2 is an ordinal sum of {([ai , bi ], Ti )}i∈ A1 such that (ai , bi ) ⊂ [0, e] \ {g(x) : x ∈ [e, 1]} for all i ∈ A1 . • U |[e,1]2 is an ordinal sum of {([c j , d j ], S j )} j ∈A2 such that (c j , d j ) ⊂ [e, 1] \ {g(x) : x ∈ [0, e]} for all j ∈ A2 .
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Remark 4.4. The sum (
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Theorem 5.1. Let U ∈ U(e) and a, b, c, d ∈ [0, 1], a ≤ b ≤ e ≤ c ≤ d be such that U |[a,b]2 is associative, increasing, with the neutral element b and U |[c,d]2 is associative, increasing, with the neutral element c. Then the set ([a, b] ∪ [c, d])2 is closed under U (cf. Fig. 6).
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5. Some new properties of uninorms
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Proof. Let U ∈ U(e). Firstly, observe that elements a, b, c, d are idempotent elements of the operation U. Since U (x, b) = U (b, x) = x and U (x, e) = U (e, x) = x for all x ∈ [a, b], then by the monotonicity of U one has U (x, y) = U (y, x) = x = min(x, y) for all y ∈ [b, e]. Similarly, one has U (x, y) = U (y, x) = x = max(x, y) for x ∈ [c, d] and y ∈ [e, c]. Now, we will show that U (x, y), U (y, x) ∈ [a, b] ∪ [c, d] for x ∈ [a, b], y ∈ [c, d]. By Lemma 2.5 one has x ≤ U (x, y) ≤ y. Let us suppose that U (x, y) ∈ / [a, b] ∪ [c, d]. Then we have two possibilities:
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• if U (x, y) ∈ (b, e], then b = U (b, U (x, y)) = U (U (b, x), y) = U (x, y), which leads to a contradiction. • if U (x, y) ∈ (e, c), then c = U (U (x, y), c) = U (x, U (y, c)) = U (x, y), again a contradiction.
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Similarly, if we suppose that U (y, x) ∈ / [a, b] ∪ [c, d], then this leads to a contradiction. As a result we obtain that U (x, y), U (y, x) ∈ [a, b] ∪ [c, d] for x ∈ [a, b], y ∈ [c, d]. Please cite this article as: P. Dryga´s, On properties of uninorms with underlying t-norm and t-conorm given as ordinal sums, Fuzzy Sets and Systems (2009), doi: 10.1016/j.fss.2009.09.017
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Fig. 5. The structure of an operation belonging to the class U (e) and locally internal on A(e).
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Fig. 6. The sets from Theorem 5.1.
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Fig. 7. The transformation of operations.
Similarly, like in paper [2], we may enlarge the set ((a, b) ∪ (c, d))2 to [0, 1]2 by a simple transformation. If ((a, b) ∪ (c, d))2 is closed under U, then we describe operations obtained by this transformation and then, by the inverse transformation, we receive description of the part of the operation U (cf. Fig. 7). Thanks to this method we should find all sets closed under U and describe the obtained operations.
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Example 5.2. Let us consider the operation U ∈ U(e) which is continuous in the open unit square, such that s ∈ [0, e), t ∈ (e, 1] are its idempotent elements considered in Theorem 3.1. Then the sets ([0, e] ∪ [e, t])2 and ([0, e] ∪ [t, 1])2 or the sets ([0, s] ∪ [e, 1])2 and ([s, e] ∪ [e, 1])2 are closed under U. Please cite this article as: P. Dryga´s, On properties of uninorms with underlying t-norm and t-conorm given as ordinal sums, Fuzzy Sets and Systems (2009), doi: 10.1016/j.fss.2009.09.017
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Conjecture 5.3. Let U ∈ U(e) and T ∗ and S ∗ be given as ordinal sums with summand intervals [ak , bk ], where k ∈ T and [cl , dl ], where l ∈ S. Then for every k ∈ T and l ∈ S the set ([ak , bk ] ∪ [cl , dl ])2 is closed under U.
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Example 5.4. Every operation U ∈ U(e) locally internal on A(e) is closed under U on the set ([ak , bk ] ∪ [cl , dl ])2 (see Theorem 4.3).
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Lemma 5.5. Let U ∈ U(e) and a, b ∈ [0, 1] be idempotent elements such that a < e < b. • If for some y ∈ (e, 1] it holds U (a, y) ≤ e (U (y, a) ≤ e, respectively), then U (a, y) = min(a, y) = a (U (y, a) = min(a, y) = a, respectively). • If for some x ∈ [0, e) it holds U (x, b) ≥ e (U (b, x) ≥ e, respectively), then U (x, b) = max(x, b) = b (U (b, x) = max(x, b) = b, respectively). Proof. Let a ∈ [0, e) be an idempotent element and y ∈ (e, 1] be such that U (a, y) ≤ e. Since a is an idempotent element, we get a = U (a, a) ≤ U (a, z) ≤ U (a, e) = a = min(a, z) for all z ∈ [a, e]. Thus U (a, y) = U (U (a, a), y) = U (a, U (a, y)) = min(a, U (a, y)) = a. As a result, U (a, y) = min(a, y). In a similar way we prove that U (y, a) = min(a, y). In the same way we may prove the second part of this lemma.
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Directly from the above we obtain Corollary 5.6. Let U ∈ U(e) and SU = max. If for some x ∈ [0, e), y ∈ (e, 1] it holds U (x, y) ≥ e (U (y, x) ≥ e, respectively), then U (x, y) = max(x, y) (U (y, x) = max(x, y), respectively).
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Corollary 5.7. Let U ∈ U(e) and TU = min. If for some x ∈ [0, e), y ∈ (e, 1] it holds U (x, y) ≤ e (U (y, x) ≤ e, respectively), then U (x, y) = min(x, y) (U (y, x) = min(x, y), respectively).
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Remark 5.8. If in Corollaries 5.6 and 5.7 we add the assumption of the commutativity of the operation U, then we obtain results similar to the ones presented in [2].
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Let us denote Ie = {x ∈ [0, e] : U (x, x) = x} and I e = {x ∈ [e, 1] : U (x, x) = x}.
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Corollary 5.9. An operation U ∈ U(e) is locally internal on the set Ie × I e ∪ I e × Ie . Lemma 5.10. Let U ∈ U(e), SU = max and TU be a continuous Archimedean pseudo-t-norm. If for some x ∈ (0, e), y ∈ (e, 1] it holds U (x, y) < e (U (y, x) < e), then U (x, y) = min(x, y) (U (y, x) = min(x, y)).
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Proof. Suppose that x < U (x, y) < e. Since TU is a continuous pseudo-t-norm, then there exists z ∈ (0, e) such that U (U (x, y), z) = x. If U (y, z) ≥ e, then by Corollary 5.6 U (x, U (y, z)) = U (x, y), i.e., U (x, y) = x, which leads to a contradiction. If U (y, z) < e, then U (x, U (y, z)) = x. This is possible only if x = 0 or U (y, z) = e which leads to a contradiction. As a result U (x, y) = x.
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Lemma 5.11. Let U ∈ U(e), SU = max and TU be a continuous Archimedean pseudo-t-norm. If for some y ∈ (e, 1] there exists x ∈ [0, e) such that U (x, y) = min(x, y) (U (y, x) = min(x, y), respectively), then U (0, y) = 0 (U (y, 0) = 0, respectively).
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Proof. If x = 0, then the statement in Lemma 5.11 is obvious. Let us now assume that x ∈ (0, e). Taking t ∈ (0, x) one has U (0, y) ≤ U (t, y) ≤ U (x, y) = x < e. By Lemma 5.10 one has U (t, y) = min(t, y). As a result 0 ≤ U (0, y) ≤ limt →0+ U (t, y) = limt →0+ min(t, y) = 0 and consequently one obtains U (0, y) = 0. Remark 5.12. Let U ∈ U(e), SU = max and TU be a continuous Archimedean pseudo-t-norm. Then by Corollary 5.6 and Lemmas 5.10 and 5.11 the operation U is locally internal on the set A(e). Moreover, by Theorem 4.3 there exists a function g that separates minimum and maximum (see Fig. 8). Additionally, since TU is a continuous Archimedean Please cite this article as: P. Dryga´s, On properties of uninorms with underlying t-norm and t-conorm given as ordinal sums, Fuzzy Sets and Systems (2009), doi: 10.1016/j.fss.2009.09.017
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pseudo-t-norm, then by Remark 4.4 one has g([0, 1]) ∩ (0, e) = ∅. As a result the function g is constant on the set (0, e) (see Fig. 8).
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Similarly to the way presented in the proofs of Lemmas 5.10 and 5.11 we may prove the following facts.
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Lemma 5.13. Let U ∈ U(e), TU = min and SU be a continuous Archimedean pseudo-t-conorm. If for some x ∈ [0, e), y ∈ (e, 1] it holds U (x, y) > e (U (y, x) > e, respectively), then U (x, y) = max(x, y) (U (y, x) = max(x, y), respectively).
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Lemma 5.14. Let U ∈ U(e), TU = min and SU be a continuous Archimedean pseudo-t-conorm. If for some x ∈ [0, e) there exists y ∈ (e, 1] such that U (x, y) = max(x, y) (U (y, x) = max(x, y), respectively), then U (x, 1) = 1 (U (1, x) = 1, respectively).
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Remark 5.15. Let U ∈ U(e), TU = min and SU be a continuous Archimedean pseudo-t-conorm. Then by Corollary 5.7 and Lemmas 5.13 and 5.14 the operation U is locally internal on the set A(e). Moreover, by Theorem 4.3 there exists a function g that separates minimum and maximum (see Fig. 9). Additionally, since SU is a continuous Archimedean pseudo-t-conorm, then by Remark 4.4 one has g([0, 1]) ∩ (e, 1) = ∅. As a result the function g is constant on the set (e, 1) (see Fig. 9). Please cite this article as: P. Dryga´s, On properties of uninorms with underlying t-norm and t-conorm given as ordinal sums, Fuzzy Sets and Systems (2009), doi: 10.1016/j.fss.2009.09.017
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6. Conclusion In this paper we have presented some properties of uninorm-like operations where the underlying operations are given as ordinal sums. If underlying operations of a uninorm are given as ordinal sums, then the Cartesian product of the union of two arbitrary intervals, each from the domain of another ordinal sum, is a set closed under the uninorm. Thus we have described the operations such that the underlying operations are the basic pseudo-t-norms and pseudo-t-conorms and one of them is idempotent. If we omit the assumption about the idempotency, then we obtain the following: Problem 6.1. What structure of a uninorm will be obtained if other basic t-norms and t-conorms, such as the product t-norm, the drastic t-norm or rotation invariant t-norms (see [15–17]) will be applied in the presented ones encompassing the considerations of this paper? References
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[1] B. De Baets, Idempotent uninorms, European J. Oper. Res. 118 (1999) 631–642. [2] B. De Baets, J. Fodor, T. Calvo, The characterization of uninorms with continuous underlying t-norms and t-conorms, Fuzzy Sets and Systems, submitted for publication. [3] A.H. Clifford, Naturally totally ordered commutative semigroups, Amer. J. Math. 76 (1954) 631–646. [4] E. Czogała, J. Drewniak, Associative monotonic operations in fuzzy set theory, Fuzzy Sets and Systems 12 (1984) 249–269. [5] J. Drewniak, P. Dryga´s, Ordered semigroups in constructions of uninorms and nullnorms, in: P. Grzegorzewski, M. Krawczak, S. Zadro˙zny, (Eds.), Issues in Soft Computing Theory and Applications, EXIT, Warszawa, 2005, pp. 147–158. [6] P. Dryga´s, Discussion of the structure of uninorms, Kybernetika 41 (2005) 213–226. [7] P. Dryga´s, Remarks about idempotent uninorms, J. Electrical Eng. 57 (2006) 92–94. [8] P. Dryga´s, On the structure of continuous uninorms, Kybernetika 43 (2007) 183–196. [9] J. Fodor, R. Yager, A. Rybalov, Structure of uninorms, Internat. J. Uncertain. Fuzziness Knowledge-Based Syst. 5 (1997) 411–427. [10] S. Gottwald, A Treatise on Many-valued Logic, Research Studies Press, Baldock, 2001. [11] S.-K. Hu, Z.-F. Li, The structure of continuous uninorms, Fuzzy Sets and Systems 124 (2001) 43–52. [12] S. Jenei, A note on the ordinal sum theorem and its consequence for the construction of triangular norm, Fuzzy Sets and Systems 126 (2002) 199–205. [13] E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer Academic Publishers, Dordrecht, 2000. [14] Y.-M. Li, Z.-K. Shi, Remarks on uninorm aggregation operators, Fuzzy Sets and Systems 114 (2000) 377–380. [15] K.C. Maes, B. De Baets, On the structure of left-continuous t-norms that have a continuous contour line, Fuzzy Sets and Systems 158 (2007) 843–860. [16] K.C. Maes, B. De Baets, Rotation-invariant t-norms: the rotation invariance property revisited, Fuzzy Sets and Systems 160 (2009) 44–51. [17] K.C. Maes, B. De Baets, Rotation-invariant t-norms: where triple rotation and rotation–annihilation meet, Fuzzy Sets and Systems 160 (2009) 1998–2016. [18] J. Martín, G. Mayor, J. Torrens, On locally internal monotonic operations, Fuzzy Sets and Systems 137 (2003) 27–42. [19] W. Sander, Associative aggregation operators, in: T. Calvo, G. Mayor, R. Mesiar (Eds.), Aggregation Operators, Physica, Heidelberg, 2002, pp. 124–158. [20] R. Yager, A. Rybalov, Uninorm aggregation operators, Fuzzy Sets and Systems 80 (1996) 111–120.
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On properties of uninorms with underlying t-norm and t-conorm given as ordinal sums Paweł Dryga´s∗
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Institute of Mathematics, University of Rzeszów, ul. Rejtana 16a, 35-959 Rzeszów, Poland
Abstract
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Uninorms as binary operations on the unit interval have been widely applied in the fuzzy set theory. This paper presents some properties of uninorm-like operations for which the underlying operations are given by ordinal sums. If the underlying operations of a uninorm are given by ordinal sums, then the Cartesian product of the union of two arbitrary intervals (one in [0, e] and the other in [e, 1], where e is the neutral element of the uninorm) is a set closed under the uninorm. When transposing such a Cartesian product to the unit square, one obtains a uninorm-like operation. As a result, we have described the uninorm-like operations for which the underlying operations are basic pseudo-t-norms and pseudo-t-conorms and one of them is idempotent. © 2009 Published by Elsevier B.V.
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Keywords: Aggregation operations; Ordinal sum; Uninorm; Associative operations
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Binary operations on the unit interval have many applications in the fuzzy set theory. Specifically, they are applied as Q1 multivalued logical connectives (cf. [13,10]). To the contrary, they give examples of aggregation operations in the unit interval (cf. [20]). For this reason it is crucial to examine and characterize such operations. Uninorms are examples of binary operations on the unit interval. They were introduced by Yager and Rybalov [20] as a generalization of triangular norms and conorms letting the neutral element e to be an arbitrary point of the unit interval. Fodor et al. [9] examined a general structure of uninorms. The frame structure of uninorms and the characterization of representable uninorms were also presented in that paper. In case of e = 1 one obtains triangular norms and in case of e = 0 one obtains triangular conorms (see [13]). When e ∈ (0, 1) one obtains operations considered in many scientific articles (see [1–9,14,18–20]). Moreover, in the papers [8,11] the characterizations of continuous uninorms are given and in the papers [1,18] the descriptions of idempotent uninorms are presented. In this paper we associate the results from the papers [2,6] and we also consider more general class of operations than uninorms, i.e., we omit the assumption of the commutativity. The main aim of this paper is to present some properties of operations for which the underlying operations are given by ordinal sums. If the underlying operations of a uninorm are given by ordinal sums, then the Cartesian product of
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1. Introduction
∗ Tel.: +48 178721160.
E-mail address: [email protected]. 0165-0114/$ - see front matter © 2009 Published by Elsevier B.V. doi:10.1016/j.fss.2009.09.017 Please cite this article as: P. Dryga´s, On properties of uninorms with underlying t-norm and t-conorm given as ordinal sums, Fuzzy Sets and Systems (2009), doi: 10.1016/j.fss.2009.09.017
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the union of two arbitrary intervals, each from domain of different ordinal sum, is a set closed under the uninorm. Consequently, we describe such uninorm-like operations where the underlying operations are basic pseudo-t-norms and pseudo-t-conorms and one of them is idempotent. In Section 2 we present some notions and descriptions of the structure of uninorms and the construction of ordinal sum of semigroups. In Section 3 we characterize the uninorms which are continuous in the open unit square. Next, in Section 4, we present the structure of idempotent uninorms given in [18]. Moreover, we deal with operations which are locally internal on some subset of their domain and we give properties of the corresponding function and its influence on the structure of underlying operations (a t-norm and a t-conorm). Finally, in Section 5, we give some new properties of uninorms. 2. Notion of uninorms We will discuss the structure of binary operations in the class
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U(e) = {U : [0, 1]2 → [0, 1] : U is increasing, associative, with the neutral element e ∈ [0, 1]}. Definition 2.1 (see Yager and Rybalov [20], Sander [19]). Let e ∈ [0, 1].
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• An operation U ∈ U(e) is called a uninorm, if it is commutative. • A uninorm T (S) is called a t-norm (a t-conorm, respectively), if it has the neutral element e = 1 (e = 0, respectively). • An operation T ∈ U(1) (S ∈ U(0), respectively) is called a pseudo-t-norm (a pseudo-t-conorm, respectively).
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Lemma 2.3 (cf. Jenei [12], Drewniak and Dryga´s [5]). Let {[ak , bk ]}k∈T be a countable family of nonoverlapping, closed and proper subintervals of [0, 1]. Let T be an operation on [0, 1] defined by ⎧ x − ak y − ak ⎨ ak + (bk − ak )Tk if x, y ∈ (ak , bk ], , T (x, y) = (1) bk − ak bk − ak ⎩ min(x, y) otherwise,
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The operation given by (1) is called the ordinal sum of {([ai , bi ], Ti )}i∈T . Similarly, we may construct the operation with the neutral element 0. More information about ordinal sums one may find in [5] or [7]. Now, we present the frame structure of operations from the class U(e).
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where Tk : [0, 1]2 → [0, 1] are associative, increasing binary operations, such that Tk ≤ min for every k ∈ T . Moreover, we assume that operations Tk have neutral element e = 1 if bk = al and Tl is with zero divisor, or bk = 1. Then T is an operation from the class U(1). Moreover, the operation T is commutative if and only if for all k ∈ T the operations Tk are commutative.
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We may use these operations to construct new t-norms or t-conorms.
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Example 2.2 (cf. Klement et al. [13]). In Table 1 we present well-known triangular norms and conorms.
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Theorem 2.4 (cf. Fodor et al. [9]). If an operation U is in the class U(e), for e ∈ (0, 1), then there exist an operation T ∈ U(1) and an operation S ∈ U(0) such that ∗ T i n [0, e]2 , U= (2) S ∗ i n [e, 1]2 , where
x , x, y ∈ [0, e], e (x − e) , x, y ∈ [e, 1]. S ∗ (x, y) = −1 (S((x), (y))), (x) = (1 − e)
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T ∗ (x, y) = −1 (T ((x), (y))), (x) =
Let us denote the operations T and S mentioned in the above theorem by TU and SU and additionally 33
A(e) = [0, e) × (e, 1] ∪ (e, 1] × [0, e). Please cite this article as: P. Dryga´s, On properties of uninorms with underlying t-norm and t-conorm given as ordinal sums, Fuzzy Sets and Systems (2009), doi: 10.1016/j.fss.2009.09.017
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Table 1 Examples of basic t-norms and t-conorms t-Norm
t-Conorm
Name of the operations
TM (x, y) = min(x, y) TP (x, y) = xy TL (x, y) = max(x + y − 1, 0) ⎧ ⎨ x if y = 1 TD (x, y) = y if x = 1 ⎩ 0 otherwise
SM (x, y) = max(x, y) SP (x, y) = x + y − xy SL (x, y) = min(x + y, 1) ⎧ ⎨ x if y = 0 SD (x, y) = y if x = 0 ⎩ 1 otherwise
Lattice operations Algebraic operations Łukasiewicz operations Drastic operations
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Fig. 1. The structure of a uninorm.
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Lemma 2.5 (cf. Fodor et al. [9]). If U : [0, 1]2 → [0, 1] is an increasing operation with the neutral element e ∈ (0, 1), then
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Furthermore, if U is associative, then U (0, 1), U (1, 0) ∈ {0, 1}. The frame structure of uninorms, by Theorem 2.4 and Lemma 2.5, may be depicted in Fig. 1.
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Theorem 2.6 (cf. Li and Shi [14]). Let e ∈ (0, 1). If T is an arbitrary t-norm and S is an arbitrary t-conorm, then formula (2) with U = min or U = max in A(e) gives a uninorm.
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Example 2.7 (cf. Fodor et al. [9]). The formula 0 if x = 0 or y = 0, xy U (x, y) = if x > 0 and y > 0, (1 − x)(1 − y) + x y
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In the next theorem and Fig. 2 there is given the characterization of uninorm-like operations which are continuous on the open unit square. Please cite this article as: P. Dryga´s, On properties of uninorms with underlying t-norm and t-conorm given as ordinal sums, Fuzzy Sets and Systems (2009), doi: 10.1016/j.fss.2009.09.017
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Fig. 3. The structure of an idempotent operation belonging to the class U ∈ U (e).
Theorem 3.1 (Dryga´s [8]). Let e ∈ (0, 1) and U ∈ U(e) be continuous in (0, 1)2 . Then one of the following two cases holds:
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(i) There exist idempotent elements s ∈ [0, e) and p ∈ [0, s] of the operation U such that U |[0,1)2 is an ordinal sum of a continuous semigroup U |[0,s]2 with the neutral element s and a continuous group U |(s,1)2 with the Archimedean property. (ii) There exist idempotent elements t ∈ (e, 1] and q ∈ [t, 1] of the operation U such that U |(0,1]2 is a dual ordinal sum of a continuous semigroup U |[t,1]2 with the neutral element t and a continuous group U |(0,t )2 with the Archimedean property.
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Theorem 4.1 (Martín et al. [18]). Let e ∈ [0, 1]. An operation U ∈ U(e) is idempotent if and only if there exists a decreasing function g : [0, 1] → [0, 1] with g(e) = e, g(x) = 0 for all x > g(0), g(x) = 1 for all x < g(1) satisfying
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inf{y : g(y) = g(x)} ≤ g 2 (x) ≤ sup{y : g(y) = g(x)}, x ∈ [0, 1]
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and such that (see also Fig. 3) ⎧ ⎨ min(x, y) if y < g(x) or (y = g(x) and x < g 2 (x)), U (x, y) = max(x, y) if y > g(x) or (y = g(x) and x > g 2 (x)), ⎩ x or y if y = g(x) and x = g 2 (x). Please cite this article as: P. Dryga´s, On properties of uninorms with underlying t-norm and t-conorm given as ordinal sums, Fuzzy Sets and Systems (2009), doi: 10.1016/j.fss.2009.09.017
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Theorem 4.3 (Dryga´s [6]). Let e ∈ (0, 1). If an operation U ∈ U(e) is locally internal on A(e), i.e., U (x, y) ∈ {x, y} for x, y ∈ A(e), then (cf. Fig. 5): • there exists a decreasing function g : [0, 1] → [0, 1] with g(e) = e, g(x) = 0 for all x > g(0), g(x) = 1 for all x < g(1), satisfying inf{y : g(y) = g(x)} ≤ g 2 (x) ≤ sup{y : g(y) = g(x)} and such that
⎧ ⎨ min(x, y) if y < g(x) or (y = g(x) and x < g 2 (x)), U (x, y) = max(x, y) if y > g(x) or (y = g(x) and x > g 2 (x)), ⎩ x or y if y = g(x) and x = g 2 (x),
(x, y) ∈ A(e).
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Example 4.2. Let e ∈ (0, 1). For the uninorms U1 and U2 presented in Fig. 4 the corresponding functions are of the following form: 1 if x ∈ [0, e), e if x ∈ [0, e], gU1 (x) = gU2 (x) = e if x ∈ [e, 1], 0 if x ∈ (e, 1].
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• U |[0,e]2 is an ordinal sum of {([ai , bi ], Ti )}i∈ A1 such that (ai , bi ) ⊂ [0, e] \ {g(x) : x ∈ [e, 1]} for all i ∈ A1 . • U |[e,1]2 is an ordinal sum of {([c j , d j ], S j )} j ∈A2 such that (c j , d j ) ⊂ [e, 1] \ {g(x) : x ∈ [0, e]} for all j ∈ A2 .
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Remark 4.4. The sum (
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Theorem 5.1. Let U ∈ U(e) and a, b, c, d ∈ [0, 1], a ≤ b ≤ e ≤ c ≤ d be such that U |[a,b]2 is associative, increasing, with the neutral element b and U |[c,d]2 is associative, increasing, with the neutral element c. Then the set ([a, b] ∪ [c, d])2 is closed under U (cf. Fig. 6).
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Proof. Let U ∈ U(e). Firstly, observe that elements a, b, c, d are idempotent elements of the operation U. Since U (x, b) = U (b, x) = x and U (x, e) = U (e, x) = x for all x ∈ [a, b], then by the monotonicity of U one has U (x, y) = U (y, x) = x = min(x, y) for all y ∈ [b, e]. Similarly, one has U (x, y) = U (y, x) = x = max(x, y) for x ∈ [c, d] and y ∈ [e, c]. Now, we will show that U (x, y), U (y, x) ∈ [a, b] ∪ [c, d] for x ∈ [a, b], y ∈ [c, d]. By Lemma 2.5 one has x ≤ U (x, y) ≤ y. Let us suppose that U (x, y) ∈ / [a, b] ∪ [c, d]. Then we have two possibilities:
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• if U (x, y) ∈ (b, e], then b = U (b, U (x, y)) = U (U (b, x), y) = U (x, y), which leads to a contradiction. • if U (x, y) ∈ (e, c), then c = U (U (x, y), c) = U (x, U (y, c)) = U (x, y), again a contradiction.
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Similarly, if we suppose that U (y, x) ∈ / [a, b] ∪ [c, d], then this leads to a contradiction. As a result we obtain that U (x, y), U (y, x) ∈ [a, b] ∪ [c, d] for x ∈ [a, b], y ∈ [c, d]. Please cite this article as: P. Dryga´s, On properties of uninorms with underlying t-norm and t-conorm given as ordinal sums, Fuzzy Sets and Systems (2009), doi: 10.1016/j.fss.2009.09.017
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Fig. 7. The transformation of operations.
Similarly, like in paper [2], we may enlarge the set ((a, b) ∪ (c, d))2 to [0, 1]2 by a simple transformation. If ((a, b) ∪ (c, d))2 is closed under U, then we describe operations obtained by this transformation and then, by the inverse transformation, we receive description of the part of the operation U (cf. Fig. 7). Thanks to this method we should find all sets closed under U and describe the obtained operations.
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Example 5.2. Let us consider the operation U ∈ U(e) which is continuous in the open unit square, such that s ∈ [0, e), t ∈ (e, 1] are its idempotent elements considered in Theorem 3.1. Then the sets ([0, e] ∪ [e, t])2 and ([0, e] ∪ [t, 1])2 or the sets ([0, s] ∪ [e, 1])2 and ([s, e] ∪ [e, 1])2 are closed under U. Please cite this article as: P. Dryga´s, On properties of uninorms with underlying t-norm and t-conorm given as ordinal sums, Fuzzy Sets and Systems (2009), doi: 10.1016/j.fss.2009.09.017
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Conjecture 5.3. Let U ∈ U(e) and T ∗ and S ∗ be given as ordinal sums with summand intervals [ak , bk ], where k ∈ T and [cl , dl ], where l ∈ S. Then for every k ∈ T and l ∈ S the set ([ak , bk ] ∪ [cl , dl ])2 is closed under U.
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Example 5.4. Every operation U ∈ U(e) locally internal on A(e) is closed under U on the set ([ak , bk ] ∪ [cl , dl ])2 (see Theorem 4.3).
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Lemma 5.5. Let U ∈ U(e) and a, b ∈ [0, 1] be idempotent elements such that a < e < b. • If for some y ∈ (e, 1] it holds U (a, y) ≤ e (U (y, a) ≤ e, respectively), then U (a, y) = min(a, y) = a (U (y, a) = min(a, y) = a, respectively). • If for some x ∈ [0, e) it holds U (x, b) ≥ e (U (b, x) ≥ e, respectively), then U (x, b) = max(x, b) = b (U (b, x) = max(x, b) = b, respectively). Proof. Let a ∈ [0, e) be an idempotent element and y ∈ (e, 1] be such that U (a, y) ≤ e. Since a is an idempotent element, we get a = U (a, a) ≤ U (a, z) ≤ U (a, e) = a = min(a, z) for all z ∈ [a, e]. Thus U (a, y) = U (U (a, a), y) = U (a, U (a, y)) = min(a, U (a, y)) = a. As a result, U (a, y) = min(a, y). In a similar way we prove that U (y, a) = min(a, y). In the same way we may prove the second part of this lemma.
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Directly from the above we obtain Corollary 5.6. Let U ∈ U(e) and SU = max. If for some x ∈ [0, e), y ∈ (e, 1] it holds U (x, y) ≥ e (U (y, x) ≥ e, respectively), then U (x, y) = max(x, y) (U (y, x) = max(x, y), respectively).
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Corollary 5.7. Let U ∈ U(e) and TU = min. If for some x ∈ [0, e), y ∈ (e, 1] it holds U (x, y) ≤ e (U (y, x) ≤ e, respectively), then U (x, y) = min(x, y) (U (y, x) = min(x, y), respectively).
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Remark 5.8. If in Corollaries 5.6 and 5.7 we add the assumption of the commutativity of the operation U, then we obtain results similar to the ones presented in [2].
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Let us denote Ie = {x ∈ [0, e] : U (x, x) = x} and I e = {x ∈ [e, 1] : U (x, x) = x}.
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Corollary 5.9. An operation U ∈ U(e) is locally internal on the set Ie × I e ∪ I e × Ie . Lemma 5.10. Let U ∈ U(e), SU = max and TU be a continuous Archimedean pseudo-t-norm. If for some x ∈ (0, e), y ∈ (e, 1] it holds U (x, y) < e (U (y, x) < e), then U (x, y) = min(x, y) (U (y, x) = min(x, y)).
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Proof. Suppose that x < U (x, y) < e. Since TU is a continuous pseudo-t-norm, then there exists z ∈ (0, e) such that U (U (x, y), z) = x. If U (y, z) ≥ e, then by Corollary 5.6 U (x, U (y, z)) = U (x, y), i.e., U (x, y) = x, which leads to a contradiction. If U (y, z) < e, then U (x, U (y, z)) = x. This is possible only if x = 0 or U (y, z) = e which leads to a contradiction. As a result U (x, y) = x.
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Lemma 5.11. Let U ∈ U(e), SU = max and TU be a continuous Archimedean pseudo-t-norm. If for some y ∈ (e, 1] there exists x ∈ [0, e) such that U (x, y) = min(x, y) (U (y, x) = min(x, y), respectively), then U (0, y) = 0 (U (y, 0) = 0, respectively).
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Proof. If x = 0, then the statement in Lemma 5.11 is obvious. Let us now assume that x ∈ (0, e). Taking t ∈ (0, x) one has U (0, y) ≤ U (t, y) ≤ U (x, y) = x < e. By Lemma 5.10 one has U (t, y) = min(t, y). As a result 0 ≤ U (0, y) ≤ limt →0+ U (t, y) = limt →0+ min(t, y) = 0 and consequently one obtains U (0, y) = 0. Remark 5.12. Let U ∈ U(e), SU = max and TU be a continuous Archimedean pseudo-t-norm. Then by Corollary 5.6 and Lemmas 5.10 and 5.11 the operation U is locally internal on the set A(e). Moreover, by Theorem 4.3 there exists a function g that separates minimum and maximum (see Fig. 8). Additionally, since TU is a continuous Archimedean Please cite this article as: P. Dryga´s, On properties of uninorms with underlying t-norm and t-conorm given as ordinal sums, Fuzzy Sets and Systems (2009), doi: 10.1016/j.fss.2009.09.017
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Similarly to the way presented in the proofs of Lemmas 5.10 and 5.11 we may prove the following facts.
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Lemma 5.13. Let U ∈ U(e), TU = min and SU be a continuous Archimedean pseudo-t-conorm. If for some x ∈ [0, e), y ∈ (e, 1] it holds U (x, y) > e (U (y, x) > e, respectively), then U (x, y) = max(x, y) (U (y, x) = max(x, y), respectively).
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Lemma 5.14. Let U ∈ U(e), TU = min and SU be a continuous Archimedean pseudo-t-conorm. If for some x ∈ [0, e) there exists y ∈ (e, 1] such that U (x, y) = max(x, y) (U (y, x) = max(x, y), respectively), then U (x, 1) = 1 (U (1, x) = 1, respectively).
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Remark 5.15. Let U ∈ U(e), TU = min and SU be a continuous Archimedean pseudo-t-conorm. Then by Corollary 5.7 and Lemmas 5.13 and 5.14 the operation U is locally internal on the set A(e). Moreover, by Theorem 4.3 there exists a function g that separates minimum and maximum (see Fig. 9). Additionally, since SU is a continuous Archimedean pseudo-t-conorm, then by Remark 4.4 one has g([0, 1]) ∩ (e, 1) = ∅. As a result the function g is constant on the set (e, 1) (see Fig. 9). Please cite this article as: P. Dryga´s, On properties of uninorms with underlying t-norm and t-conorm given as ordinal sums, Fuzzy Sets and Systems (2009), doi: 10.1016/j.fss.2009.09.017
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6. Conclusion In this paper we have presented some properties of uninorm-like operations where the underlying operations are given as ordinal sums. If underlying operations of a uninorm are given as ordinal sums, then the Cartesian product of the union of two arbitrary intervals, each from the domain of another ordinal sum, is a set closed under the uninorm. Thus we have described the operations such that the underlying operations are the basic pseudo-t-norms and pseudo-t-conorms and one of them is idempotent. If we omit the assumption about the idempotency, then we obtain the following: Problem 6.1. What structure of a uninorm will be obtained if other basic t-norms and t-conorms, such as the product t-norm, the drastic t-norm or rotation invariant t-norms (see [15–17]) will be applied in the presented ones encompassing the considerations of this paper? References
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