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\({\mathscr {Q}}{\mathscr {L}}\)-(operators) implications derived from quasi-overlap (quasi-grouping) functions and negations on bounded lattices

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Abstract

Fuzzy implications, as a generalization of the classical implication, are not only required in fuzzy logic systems and fuzzy control but also have an important effect on solving fuzzy relational equations, fuzzy mathematical morphology, image processing and etc. Therefore, it is necessary for us to investigate multiple types of fuzzy implications on different truth values sets. In this paper, we devote to propose the \({\mathscr {Q}}{\mathscr {L}}\)-(operators) implications derived from quasi-overlap (quasi-grouping) functions and negations on bounded lattices. Firstly, we exactly investigate some desirable properties of \({\mathscr {Q}}{\mathscr {L}}\)-operators. Afterwards, we provide a necessary and sufficient condition along with a sufficient condition for the \({\mathscr {Q}}{\mathscr {L}}\)-operator to be a \({\mathscr {Q}}{\mathscr {L}}\)-implication and study various prime properties of \({\mathscr {Q}}{\mathscr {L}}\)-implications. Moreover, we show the relationship between \({\mathscr {Q}}{\mathscr {L}}\)-implications and L-automorphisms on bounded lattices. Finally, we consider the \({\mathscr {Q}}{\mathscr {L}}\)-implications to analyze their intersection with \({\mathscr {I}}_{{\mathscr {G}},{\mathscr {N}}}\)-implications derived from quasi-grouping functions and negations on bounded lattices.

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Notes

  1. An element p in a bounded lattice L is called prime iff \(x\wedge y\le _{L} p\) always implies \(x\le _{L} p\) or \(y\le _{L} p\) Gierz et al. (2003).

  2. A binary operator \(S:L\times L\longrightarrow L\) is a t-conorm on the bounded lattice L if it satisfies the four properties: commutative, associative, increasing and \(0_L\) is the neutral element. Furthermore, S is said to be positive if \(S(x, y)= 1_L\) then either \(x = 1_L\) or \(y = 1_L\)Bedregal et al. (2013).

  3. An element p in a bounded lattice L is called co-prime iff \(p\le _{L}x\vee y\) always implies \(p\le _{L}x\) or \(p\le _{L}y\) Gierz et al. (2003).

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Acknowledgements

The authors would like to express their sincerely thanks to the Editors and anonymous reviewers for their most valuable comments and suggestions in improving this paper greatly.

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  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, People’s Republic of China

    Yun Song & Junsheng Qiao

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  1. Yun Song
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This work was supported by the National Natural Science Foundation of China (62166037 and 11901465), the Science and Technology Program of Gansu Province (20JR10RA101).

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Song, Y., Qiao, J. \({\mathscr {Q}}{\mathscr {L}}\)-(operators) implications derived from quasi-overlap (quasi-grouping) functions and negations on bounded lattices. Comp. Appl. Math. 42, 239 (2023). https://doi.org/10.1007/s40314-023-02367-x

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