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On the characterization of the structure of distributive uninorms

Wenwen Zong, Yong Su, Hua-Wen Liu

Abstract

This paper focuses on distributive uninorms, which induce structures of commutative ordered semirings. We will show that the second uninorm must be locally internal on $A(e)$, and will present a complete characterization of the structure of such uninorms.

On the characterization of the structure of distributive uninorms

Abstract

This paper focuses on distributive uninorms, which induce structures of commutative ordered semirings. We will show that the second uninorm must be locally internal on , and will present a complete characterization of the structure of such uninorms.

Paper Structure

This paper contains 5 sections, 7 theorems, 6 equations, 2 figures.

Table of Contents

  1. Introduction, basic notions and results
  2. Main results
  3. The case $e_1 = e_2$
  4. The case $e_1 > e_2$
  5. The case $e_1 < e_2$

Key Result

Theorem 1

SuLiuetal18 Let $U_1$ and $U_2$ be two uninorms with the same neutral element $e$. Then $U_1$ is distributive over $U_2$ if and only if $U_2$ is idempotentA uninorm $U$ is idempotent if $U(x, x) = x$ for all $x \in [0, 1]$. and $U_1(x,y) = U_2(x,y) \in \{x, y\}$ for all $(x,y) \in A(e)$.

Figures (2)

  • Figure 1: The structure of uninorms $U_1$ and $U_2$ when $e_2<e_1$.
  • Figure 2: The structure of uninorms $U_1$ and $U_2$ when $e_1<e_2$.

Theorems & Definitions (10)

  • Theorem 1
  • Proposition 1
  • proof
  • Corollary 1
  • Theorem 2
  • proof
  • Remark 1
  • Proposition 2
  • Corollary 2
  • Theorem 3