Academic Background and Problem Statement

In the field of fuzzy sets (Fuzzy Sets, FSs), handling uncertainty is one of the core challenges. Fuzzy sets were first introduced by Zadeh in 1965 and quickly became a hot topic in theoretical and applied research. With the deepening of research, an extended form of fuzzy sets—Orthopair Fuzzy Sets (OFSs)—emerged. OFSs introduce orthopairs (i.e., membership and non-membership degrees) to more comprehensively describe uncertain information. Yager first defined OFSs in 2013 and proposed the concept of q-Rung Orthopair Fuzzy Sets (q-ROFSs). Subsequently, Gao and Zhang further introduced Linear Orthopair Fuzzy Sets (LOFs) and their symmetrical form—Symmetrical Linear Orthopair Fuzzy Sets (SLOFs) in 2021.

Although t-norms and t-conorms have been widely applied in q-ROFSs, their application in SLOFs has not been thoroughly explored. T-norms and t-conorms are fundamental operations in fuzzy set theory and play a crucial role in Multiple-Criteria Decision-Making (MCDM). Therefore, this paper aims to fill this research gap by extending and simulating relevant results in Intuitionistic Fuzzy Sets (IFSs) to explore t-norms, t-conorms, and their applications in MCDM within SLOFs.

Source and Author Information

This paper is co-authored by Shan Gao, Xianyong Zhang, and Zhiwen Mo, who are affiliated with the School of Mathematical Sciences at Sichuan Normal University, the Science Department at Taiyuan Institute of Technology, and the Laurent Mathematics Center at Sichuan Normal University, respectively. The paper was accepted by the journal Cognitive Computation on March 5, 2025, and published in the same year. This journal is a well-known academic publication under Springer, focusing on research in cognitive computing and its applications.

Research Process and Main Content

1. Research Process

The research process of this paper is divided into the following steps:

a) Theoretical Foundation and Definitions

First, the authors review the basic definitions and constraints of OFSs, q-ROFSs, LOFs, and SLOFs. Specifically, SLOFs are categorized into three types: ω ≥ 1, 0.5 < ω < 1, and 0 < ω ≤ 0.5. Then, the authors introduce the basic definitions of t-norms and t-conorms and explore their applications in IFSs.

b) Construction of t-Norms and t-Conorms in SLOFs

Based on t-norms and t-conorms in IFSs, the authors propose axiomatic definitions of t-norms and t-conorms in SLOFs. By simulating and extending relevant results in IFSs, they construct t-norms and t-conorms in SLOFs. Specifically, the authors first present the general properties of t-norms and t-conorms in SLOFs and then propose several special t-norms and t-conorms through specific construction methods.

c) Addition and Scalar Multiplication Operations

Based on the constructed t-norms and t-conorms, the authors define addition and scalar multiplication operations in SLOFs and explore the properties of these operations. These operations provide the foundation for subsequent aggregation operations.

d) Design of MCDM Method

Finally, the authors design an MCDM method based on SLOFs, which combines addition, scalar multiplication, and aggregation operations. By comparing it with the corresponding method in q-ROFSs, they validate its high reliability.

2. Main Results

a) t-Norms and t-Conorms in SLOFs

By simulating and extending t-norms and t-conorms in IFSs, the authors successfully construct t-norms and t-conorms in SLOFs. Specifically, they propose t-norms and t-conorms for three types of SLOFs and prove their rationality. These results provide a theoretical foundation for operations in SLOFs.

b) Addition and Scalar Multiplication Operations

Based on the constructed t-norms and t-conorms, the authors define addition and scalar multiplication operations in SLOFs and demonstrate that these operations possess favorable mathematical properties. These operations support subsequent aggregation operations.

c) MCDM Method

The authors design an MCDM method based on SLOFs and validate its high reliability through two practical cases. Compared to the corresponding method in q-ROFSs, this method exhibits greater flexibility and accuracy in handling cognitive information.

Conclusions and Significance

By extending and simulating t-norms and t-conorms in IFSs, this paper successfully constructs t-norms and t-conorms in SLOFs and defines addition and scalar multiplication operations. These operations provide the foundation for aggregation operations in SLOFs and further design an MCDM method based on SLOFs. This method demonstrates high reliability in practical applications, offering new tools for handling cognitive information.

Highlights of the Research

1. Filling the Research Gap: This paper introduces t-norms and t-conorms in SLOFs for the first time, filling a gap in this field.
2. Operation Construction: By simulating and extending relevant results in IFSs, the authors successfully construct t-norms, t-conorms, and addition and scalar multiplication operations in SLOFs.
3. MCDM Method: An MCDM method based on SLOFs is designed, and its high reliability is validated through practical cases.

Other Valuable Information

This research not only enriches the theoretical system of OFSs but also provides new methods for cognitive computing and its applications. In particular, the MCDM method based on SLOFs has broad application prospects in handling complex decision-making problems. Future research could further explore the applications of SLOFs in other fields, such as artificial intelligence and data mining.