Incomparability with respect to the triangular order

Aşıcı, Emel; Karaçal, Funda

doi:10.14736/kyb-2016-1-0015

Cited by 6 publications

(21 citation statements)

References 16 publications

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“…(ii) Define a mapping U : L 2 1 → L 1 by Table 6 such that U is constructed using the equality (2). Then, by Theorem 3.5, U is a uninorm on L with a neutral element e. Theorem 3.9.…”

Section: Uninorms With Fixed Underlying T-norms and T-conormsmentioning

confidence: 99%

“…These operators have been extensively used in many application in fuzzy set theory, fuzzy logics, multicriteria decision support and several branches of information sciences. For more details on tnorms, we refer to [1,2,17]. Uninorms were introduced in [22] and further investigated in [23] by Yager and Rybalov and in [14] by Fodor, Yager and Rybalov, which are also generalizations of t-norms and t-conorms.…”

Section: Introductionmentioning

confidence: 99%

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Construction of uninorms on bounded lattices

Çaylı¹,

Karaçal²

2017

Kybernetika

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“…(ii) Define a mapping U : L 2 1 → L 1 by Table 6 such that U is constructed using the equality (2). Then, by Theorem 3.5, U is a uninorm on L with a neutral element e. Theorem 3.9.…”

Section: Uninorms With Fixed Underlying T-norms and T-conormsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Construction of uninorms on bounded lattices

Çaylı¹,

Karaçal²

2017

Kybernetika

Self Cite

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“…All cases when both uninorm lay in any one of the most usual classes of uninorms are analyzed, characterizing all solutions of the migrativity equation for some possible combinations. Nullnorms, uninorms and t-norms were also studied by many other authors [1,2,3,4,6,7,8,17,18]. In the present paper, we introduce the migrativity of triangular norms over nullnorms and over uninorms.…”

mentioning

confidence: 87%

On the \alpha-migrativity of t-norms and t-conorms over nullnorms and uninorms

Aşıcı¹

2018

ntmsci

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In this paper the notions of α-migrative triangular norms over a fixed nullnorm and a fixed uninorm are introduced and studied. All solutions of the migrativity equation for all possible combinations of uninorms and nullnorms are analyzed and characterized. Similar study is done for triangular conorms. 154Emel Asici: On the α-migrativity of t-norms and t-conorms over nullnorms and uninorms Emel Asici: On the α-migrativity of t-norms and t-conorms over nullnorms and uninorms Now, we will introduce the definition of migrativity of a t-conorm S over a nullnorm F. Definition 7. Given a nullnorm F and α ∈]0, 1[, a t-conorm S is called α-migrative over F or (α, F)-migrative if S(F(α, x), y) = S(x, F(α, y)) f or all x, y ∈ [0, 1].(3)Since the extreme values of a correspond to the well known cases of t-norms and t-conorms, we will only deal with nullnorms with absorbing element a ∈]0, 1[. Lemma 2. Consider α ∈]0, 1[. Let S be a t-conorm, F be a nullnorm with absorbing element a. Then S is not α-migrative over F.It leads a contradiction that S is not α-migrative over F.It leads a contradiction that S is not α-migrative over F.Similarly to the case of nullnorms we want to study the migrativity of t-norms over uninorms. Definition 8. Given a uninorm U and αSince the extreme values of e correspond to the well known cases of t-norms and t-conorms, we will only deal with uninorms with neutral element e ∈]0, 1[. Lemma 3.Consider T be a t-norm and U be a uninorm with neutral element e. Then, T is e-migrative over U.Proof. T (U(e, x), y) = T (x, y) = T (x,U(e, y)) for all x, y ∈ [0, 1].Lemma 4. Consider α ∈]0, e[. Let T be a t-norm, U be a uninorm with neutral element e. Then T is not α-migrative over U.Proof. Since 0 < α, we have that U(0, 0) = 0 < U(α, 0). If T is α-migrative over U, then we haveLemma 5. Consider α ∈]e, 1[. Let T be a t-norm, U be a uninorm with neutral element e. Then T is not α-migrative over U. Proof. Since e < α, we have 1 = U(e, 1) < U(α, 1) c 2018 BISKA Bilisim Technology NTMSCI 6, No. 1, 153-158 (2018) / www.ntmsci.com 157by the monotonicity of U. So, it is obtained that U(α, 1) = 1. If T is α-migrative over U, then we have T (e,U(α, 1)) = T (e, 1) = e < α = T (α, 1) = T (U(α, e), 1) contradiction. So, T is not α-migrative over U.Definition 9. Given a uninorm U and α ∈]0, 1[, a t-conorm S is called α-migrative over U or (α,U)-migrative if S (U(α, x), y) = S(x,U(α, y)) f or all x, y ∈ [0, 1].Since the extreme values of e correspond to the well known cases of t-norms and t-conorms, we will only deal with uninorms with neutral element e ∈]0, 1[. Lemma 6. Consider S be a t-conorm and U be a uninorm with neutral element e. Then, S is e-migrative over U. Proof. S(U(e, x), y) = S(x, y) = S(x,U(e, y)) for all x, y ∈ [0, 1]. Lemma 7. Consider α ∈]0, e[. Let S be a t-conorm, U be a uninorm with neutral element e. Then S is not α-migrative over U. Proof. Since α < e, we have that U(α, 0) < U(e, 0) = 0 by the monotonicity of U. So, it is obtained that U(α, 0) = 0. If S is α-migrative over U, then we have S(U(α, e)...

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“…An operation T : L 2 → L is called a t-norm on L if it is commutative, associative, increasing with respect to both variables and has the neutral element 1 such that T (x, 1) = x, for all x ∈ L. Definition 2.5. (Aşıcı and Karaçal [1], Aşıcı [2,3], Ç aylı and Karaçal [8]) Let (L, ≤ , 0, 1) be a bounded lattice. An operation S : L 2 → L is called a t-conorm on L if it is commutative, associative, increasing with respect to both variables and has the neutral element 0 such that S (x, 0) = x, for all x ∈ L.…”

Section: Constructions Of T-norms and T-conormsmentioning

confidence: 99%

“…Theorem 2.10. (Drossos and Navara [13], Drossos [14]) Let (L, ≤, 0, 1) be a bounded lattice, cl : L → L be a closure operator on L and int : L → L be an interior operator on L. Then the functions T (1) : L 2 → L and S (1) : L 2 → L are, respectively, a t-norm and a t-conorm on L, where…”

Section: Consider the Lattice Lmentioning

confidence: 99%