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Open AccessArticle

Sufficient Conditions for Triangular Norms Preserving ⊗-Convexity

1,2,* and 3
School of Science, Xi’an University of Posts and Telecommunications, Xi’an 710121, China
Shaanxi Key Laboratory of Network Data Analysis and Intelligent Processing, Xi’an University of Posts and Telecommunications, Xi’an 710121, China
School of Statistics, Xi’an University of Finance and Economics, Xi’an 710100, China
Author to whom correspondence should be addressed.
Symmetry 2018, 10(12), 729;
Received: 5 November 2018 / Revised: 25 November 2018 / Accepted: 30 November 2018 / Published: 7 December 2018
(This article belongs to the Special Issue Discrete Mathematics and Symmetry)
PDF [236 KB, uploaded 7 December 2018]


The convexity in triangular norm (for short, ⊗−convexity) is a generalization of Zadeh’s quasiconvexity. The aggregation of two ⊗−convex sets is under the aggregation operator ⊗ is also ⊗−convex, but the aggregation operator ⊗ is not unique. To solve it in complexity, in the present paper, we give some sufficient conditions for aggregation operators preserve ⊗−convexity. In particular, when aggregation operators are triangular norms, we have that several results such as arbitrary triangular norm preserve D convexity and a convexity on bounded lattices, M preserves H convexity in the real unite interval [ 0 , 1 ] . View Full-Text
Keywords: aggregation operator; triangular norm; ⊗-convex set aggregation operator; triangular norm; ⊗-convex set

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Li, L.; Luo, Q. Sufficient Conditions for Triangular Norms Preserving ⊗-Convexity. Symmetry 2018, 10, 729.

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