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Binary relations induced from quasi-overlap functions and quasi-grouping functions on a bounded lattice

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Abstract

Binary relations induced from aggregation operations on a bounded lattice have received more and more attention. In this paper, we investigate binary relations induced from a quasi-overlap function and a quasi-grouping function on a bounded lattice L, respectively. At first, we provide a new approach to generate binary relations by a lattice-valued quasi-overlap function and a lattice-valued quasi-grouping function. In this new approach, we get rid of the restriction of the Intermediate Value Theorem, which is the main tool in the case \(L=[0,1]\). Then we discuss the connections between the induced binary relations and the natural order on L. Finally, we demonstrate the conditions under which the binary relations can become a reflexive, anti-symmetric, transitive relation as well as a partial order. In particular, we explore the binary relations induced from two classes of special quasi-overlap (resp. grouping) functions, which are generated by a quasi-pseudo-automorphism and a \(0_L\), \(1_L\)-aggregation function, respectively.

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Notes

  1. A binary operator T: \(L\times L\longrightarrow L\) is called a t-norm (Bedregal et al. 2012) on a bounded lattice L if it is commutative, associate, increasing and \(1_L\) is the neutral element. In addition, a t-norm T is said to be positive provided that \(T(x,y)=0_L\) implies \(x=0_L\) or \(y=0_L\).

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Acknowledgements

The authors are thankful to the anonymous referees for their constructive comments and suggestions.

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Authors and Affiliations

  1. Beijing Key Laboratory on MCAACI, School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 102488, China

    Yan Sun & Bin Pang

  2. School of Applied Technology, China University of Labor Relations, Beijing, 100048, China

    Shao-Yu Zhang

Authors
  1. Yan Sun
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  2. Bin Pang
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  3. Shao-Yu Zhang
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Correspondence to Bin Pang.

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Communicated by Regivan Hugo Nunes Santiago.

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This work is supported by the National Natural Science Foundation of China (Nos. 12071033, 11871097) and Beijing Institute of Technology Science and Technology Innovation Plan Cultivation Project (No. 2021CX01030)

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Sun, Y., Pang, B. & Zhang, SY. Binary relations induced from quasi-overlap functions and quasi-grouping functions on a bounded lattice. Comp. Appl. Math. 41, 340 (2022). https://doi.org/10.1007/s40314-022-02048-1

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