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\(R_{\mathscr {O}}\)-implications induced from \(C_L\)-overlap functions on complete lattices

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Abstract

Recently, the concept of overlap functions on complete lattices has been introduced by extending the truth values set from the unit closed interval to complete lattices. On the other hand, the residual implications induced from commonly used aggregation functions (see, e.g., t-norms, pseudo-t-norms, uninorms, semi-uninorms and pseudo-uninorms), as a natural research topic of these commonly used aggregation functions in the case of lattice values, play a vital role in many-valued logic. In this paper, we consider the residual implications derived from the so-called \(C_L\)-overlap functions on complete lattices which are the weak form of overlap functions on complete lattices. To be precise, firstly, we give the notion of \(R_{\mathscr {O}}\)-implications which are residual implications induced from the \(C_L\)-overlap functions on complete lattices and give some basic properties of them. Secondly, we focus on the conditions under which \(R_{\mathscr {O}}\)-implications can satisfy the certain algebraic properties possessed by implications on complete lattices. Finally, we give a one-to-one correspondence between different families of certain implications on complete lattices and the family of \(C_L\)-overlap functions on complete lattices.

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Notes

  1. An element p in a complete lattice L is called prime iff \(x\wedge y\le p\) always implies that \(x\le p\) or \(y\le p\) (Gierz et al. 1980, 2003).

  2. A binary operator \(T:L\times L\longrightarrow L\) is a t-norm if it is commutative, associative, increasing and \(1_L\) is the neutral element. In addition, T is said to be continuous if it is left and right continuous at the same time and positive if \(T(x, y)= 0_L\) then either \(x = 0_L\) or \(y = 0_L\) (Zhang 2005).

References

  • Asmus TC, Dimuro GP, Bedregal B, Sanz JA, Pereira S Jr, Bustince H (2020) General interval-valued overlap functions and interval-valued overlap indices. Inf Sci 527:27–50

    Article  MathSciNet  MATH  Google Scholar 

  • Asmus TC, Sanz JA, Dimuro GP, Bedregal B, Fernández J, Bustince H (2021) \(N\)-Dimensional admissibly ordered interval-valued overlap functions and its influence in interval-valued fuzzy-rule-based classification systems. IEEE Trans Fuzzy Syst 30:1060–1072

    Article  Google Scholar 

  • Asmus TC, Dimuro GP, Bedregal B, Sanz JA, Mesiar R, Bustince H (2021) Towards interval uncertainty propagation control in bivariate aggregation processes and the introduction of width-limited interval-valued overlap functions. Fuzzy Sets Syst. https://doi.org/10.1016/j.fss.2021.09.005

  • Baczyński M, Beliakov G, Bustince H, Pradera A (2013) Advances in fuzzy implication functions. Springer, Berlin

    Book  MATH  Google Scholar 

  • Bedregal B, Bustince H, Palmeira E, Dimuro G, Fernandez J (2017) Generalized interval-valued OWA operators with interval weights derived from interval-valued overlap functions. Int J Approx Reason 90:1–16

    Article  MathSciNet  MATH  Google Scholar 

  • Bedregal B, Dimuro GP, Bustince H, Barrenechea E (2013) New results on overlap and grouping functions. Inf Sci 249:148–170

    Article  MathSciNet  MATH  Google Scholar 

  • Beliakov G, Pradera A, Calvo T (2007) Aggregation functions: a guide for practitioners. Springer, Berlin

    MATH  Google Scholar 

  • Bustince H, Fernandez J, Mesiar R, Montero J, Orduna R (2010) Overlap functions. Nonlinear Anal 72:1488–1499

    Article  MathSciNet  MATH  Google Scholar 

  • Bustince H, Pagola M, Mesiar R, Hüllermeier E, Herrera F (2012) Grouping, overlaps, and generalized bientropic functions for fuzzy modeling of pairwise comparisons. IEEE Trans Fuzzy Syst 20:405–415

    Article  Google Scholar 

  • Cao M, Hu BQ, Qiao J (2018) On interval \(({\mathbb{G}},{\mathbb{N}})\)-implications and \(({\mathbb{O}},{\mathbb{G}},{\mathbb{N}})\)-implications derived from interval overlap and grouping functions. Int J Approx Reason 100:135–160

    Article  MathSciNet  MATH  Google Scholar 

  • De Miguel L, Gómez D, Rodríguez JT, Montero J, Bustince H, Dimuro GP, Sanz JA (2019) General overlap functions. Fuzzy Sets Syst 372:81–96

    Article  MathSciNet  MATH  Google Scholar 

  • Dilworth RP, Ward N (1939) Residuated lattices. Trans Am Math Soc 45:335–354

    Article  MathSciNet  MATH  Google Scholar 

  • Dimuro GP, Bedregal B (2014) Archimedean overlap functions: The ordinal sum and the cancellation, idempotency and limiting properties. Fuzzy Sets Syst 252:39–54

    Article  MathSciNet  MATH  Google Scholar 

  • Dimuro GP, Bedregal B (2015) On residual implications derived from overlap functions. Inf Sci 312:78–88

    Article  MathSciNet  MATH  Google Scholar 

  • Dimuro GP, Bedregal B (2015) On the laws of contraposition for residual implications derived from overlap functions. In: 2015 IEEE international conference on fuzzy systems (FUZZ-IEEE), IEEE

  • Dimuro GP, Bedregal B, Bustince H, Jurio A, Baczyński M, Miś K (2017) \(QL\)-operations and \(QL\)-implication functions constructed from tuples \((O, G, N)\) and the generation of fuzzy subsethood and entropy measures. Int J Approx Reason 82:170–192

    Article  MathSciNet  MATH  Google Scholar 

  • Dimuro GP, Bedregal B, Fernandez J, Sesma-Sara M, Pintor JM, Bustince H (2019) The law of \(O\)-conditionality for fuzzy implications constructed from overlap and grouping functions. Int J Approx Reason 105:27–48

    Article  MathSciNet  MATH  Google Scholar 

  • Dimuro GP, Bedregal B, Bustince H, Asiáin MJ, Mesiar R (2016) On additive generators of overlap functions. Fuzzy Sets Syst 287:76–96

    Article  MathSciNet  MATH  Google Scholar 

  • Elkano M, Galar M, Sanz J, Bustince H (2016) Fuzzy Rule-Based Classification Systems for multi-class problems using binary decomposition strategies: On the influence of n-dimensional overlap functions in the Fuzzy Reasoning Method. Inf Sci 332:94–114

    Article  Google Scholar 

  • Elkano M, Galar M, Sanz J, Fernández A, Barrenechea E, Herrera F, Bustince H (2015) Enhancing multi-class classification in FARC-HD fuzzy classifier: On the synergy between n-dimensional overlap functions and decomposition strategies. IEEE Trans Fuzzy Syst 23:1562–1580

    Article  Google Scholar 

  • Elkano M, Galar M, Sanz JA, Schiavo PF, Pereira S Jr, Dimuro GP, Borges EN, Bustince H (2018) Consensus via penalty functions for decision making in ensembles in fuzzy rule-based classification systems. Appl Soft Comput 67:728–740

    Article  Google Scholar 

  • Gierz G, Hofmann KH, Keimel K, Lawson JD, Mislove MW, Scott DS (1980) A compendium of continuous lattices. Springer, Berlin

    Book  MATH  Google Scholar 

  • Gierz G, Hofmann KH, Keimel K, Lawson JD, Mislove MW, Scott DS (2003) Continuous lattices and domains. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Gómez D, Montero J (2004) A discussion on aggregation functions. Kybernetika 40:107–120

    MathSciNet  MATH  Google Scholar 

  • Gómez D, Rodríguez JT, Montero J, Bustince H, Barrenechea E (2016) n-dimensional overlap functions. Fuzzy Sets Syst 287:57–75

    Article  MathSciNet  MATH  Google Scholar 

  • Gómez D, Rodríguez JT, Yáñez J, Montero J (2016) A new modularity measure for Fuzzy Community detection problems based on overlap and grouping functions. Int J Approx Reason 74:88–107

    Article  MathSciNet  MATH  Google Scholar 

  • Jurio A, Bustince H, Pagola M, Pradera A, Yager R (2013) Some properties of overlap and grouping functions and their application to image thresholding. Fuzzy Sets Syst 229:69–90

    Article  MathSciNet  MATH  Google Scholar 

  • Liu HW (2012) Semi-uninorms and implications on a complete lattice. Fuzzy Sets Syst 191:72–82

    Article  MathSciNet  MATH  Google Scholar 

  • Liu H, Zhao B (2019) On distributivity equations of implications over overlap functions and contrapositive symmetry equations of implications. J Intell Fuzzy Syst 36:283–294

    Article  Google Scholar 

  • Lucca G, Dimuro GP, Fernández J, Bustince H, Bedregal B, Sanz JA (2018) Improving the performance of fuzzy rule-based classification systems based on a non-averaging generalization of CC-integrals named \(C_{F_1F_2}\)-integrals. IEEE Trans Fuzzy Syst 27:124–134

    Article  Google Scholar 

  • Lucca G, Sanz JA, Dimuro GP, Bedregal B, Asiáin MJ, Elkano M, Bustince H (2017) CC-integrals: Choquet-like Copula-based aggregation functions and its application in fuzzy rule-based classification systems. Knowl-Based Syst 119:32–43

    Article  Google Scholar 

  • Lucca G, Sanz JA, Dimuro GP, Bedregal B, Bustince H, Mesiar R (2018) \(C_F\)-integrals: A new family of pre-aggregation functions with application to fuzzy rule-based classification systems. Inf Sci 435:94–110

    Article  MATH  Google Scholar 

  • Paiva R, Bedregal B, Santiago R, Vieira T (2021) Residuated implications derived from quasi-overlap functions on lattices. Int J Approx Reason 134:95–110

    Article  MathSciNet  MATH  Google Scholar 

  • Paiva R, Palmeira E, Santiago R, Bedregal B (2021) Lattice-valued overlap and quasi-overlap functions. Inf Sci 562:180–199

    Article  MathSciNet  Google Scholar 

  • Palmeira ES, Bedregal B, Bustince H, Paternain D, Miguel LD (2018) Application of two different methods for extending lattice-valued restricted equivalence functions used for constructing similarity measures on \(L\)-fuzzy sets. Inf Sci 441:95–112

    Article  MathSciNet  MATH  Google Scholar 

  • Paternain D, Bustince H, Pagola M, Sussner P, Kolesárová A, Mesiar R (2016) Capacities and overlap indexes with an application in fuzzy rule-based classification systems. Fuzzy Sets Syst 305:70–94

    Article  MathSciNet  MATH  Google Scholar 

  • Qiao J (2021) Overlap and grouping functions on complete lattices. Inf Sci 542:406–424

    Article  MathSciNet  MATH  Google Scholar 

  • Qiao J (2019) On binary relations induced from overlap and grouping functions. Int J Approx Reason 106:155–171

    Article  MathSciNet  MATH  Google Scholar 

  • Qiao J, Hu BQ (2017) On interval additive generators of interval overlap functions and interval grouping functions. Fuzzy Sets Syst 323:19–55

    Article  MathSciNet  MATH  Google Scholar 

  • Su Y, Liu HW (2015) Characterizations of residual coimplications of pseudo-uninorms on a complete lattice. Fuzzy Sets Syst 261:44–59

    Article  MathSciNet  MATH  Google Scholar 

  • Su Y, Liu HW, Pedrycz W (2017) Coimplications derived from pseudo-uninorms on a complete lattice. Int J Approx Reason 90:107–119

    Article  MathSciNet  MATH  Google Scholar 

  • Su Y, Wang ZD (2013) Pseudo-uninorms and coimplications on a complete lattice. Fuzzy Sets Syst 224:53–62

    Article  MathSciNet  MATH  Google Scholar 

  • Sun F, Wang X, Qu X, Shu Q, Zhang X (2019) Residual operations of monotone binary operations over complete lattices. Int J Approx Reason 110:127–144

    Article  MathSciNet  MATH  Google Scholar 

  • Ti L, Zhou H (2018) On \((O, N)\)-coimplications derived from overlap functions and fuzzy negations. J Intell Fuzzy Syst 34:3993–4007

    Article  Google Scholar 

  • Wang ZD (2006) Generating pseudo-t-norms and implication operators. Fuzzy Sets Syst 157:398–410

    Article  MathSciNet  MATH  Google Scholar 

  • Wang ZD (2016) Left (right) semi-uninorms and coimplications on a complete lattice. Fuzzy Sets Syst 287:227–239

    Article  MathSciNet  MATH  Google Scholar 

  • Wang ZD, Fang JX (2007) On the direct decomposability of pseudo-t-norms, t-norms and implication operators on product lattices. Fuzzy Sets Syst 158:2494–2503

    Article  MathSciNet  MATH  Google Scholar 

  • Wang ZD, Fang JX (2008) Residual coimplicators of left and right uninorms on a complete lattice. Fuzzy Sets Syst 160:2086–2096

    Article  MathSciNet  MATH  Google Scholar 

  • Wang ZD, Fang JX (2009) Residual operations of left and right uninorms on a complete lattice. Fuzzy Sets Syst 160:22–31

    Article  MathSciNet  MATH  Google Scholar 

  • Wang YM, Liu HW (2019) The modularity condition for overlap and grouping functions. Fuzzy Sets Syst 372:97–110

    Article  MathSciNet  MATH  Google Scholar 

  • Wang ZD, Niu MX, Hao XY (2015) Constructions of coimplications and left (right) semi-uninorms on a complete lattice. Inf Sci 317:181–195

    Article  MathSciNet  MATH  Google Scholar 

  • Wang ZD, Wang Y, Niu MX, Hao XY (2017) Constructing strict left (right)-disjunctive left (right) semi-uninorms and coimplications satisfying the order property. Fuzzy Sets Syst 323:79–93

    Article  MathSciNet  MATH  Google Scholar 

  • Wang ZD, Yu YD (2002) Pseudo-t-norms and implication operators on a complete Brouwerian lattice. Fuzzy Sets Syst 132:113–124

    Article  MathSciNet  MATH  Google Scholar 

  • Wang ZD, Yu YD (2003) Pseudo-t-norms and implication operators: direct products and direct product decompositions. Fuzzy Sets Syst 139:673–683

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang D (2005) Triangular norms on partially ordered sets. Fuzzy Sets Syst 153:195–209

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang T, Qin F (2020) On distributive laws between 2-uninorms and overlap (grouping) functions. Int J Approx Reason 119:353–372

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang T, Qin F, Li W (2021) On the distributivity equations between uni-nullnorms and overlap (grouping) functions. Fuzzy Sets Syst 403:56–77

    Article  MathSciNet  MATH  Google Scholar 

  • Zhou H, Yan X (2021) Migrativity properties of overlap functions over uninorms. Fuzzy Sets Syst 403:10–37

    Article  MathSciNet  MATH  Google Scholar 

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Funding

This work was supported by the National Natural Science Foundation of China (62166037 and 11901465), the Science and Technology Program of Gansu Province (20JR10RA101), the China Postdoctoral Science Foundation (2021M692561), the Scientific Research Fund for Young Teachers of Northwest Normal University (NWNU-LKQN-18-28) and the Doctoral Research Fund of Northwest Normal University (6014/0002020202).

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

    Junsheng Qiao

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  1. Junsheng Qiao
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Qiao, J. \(R_{\mathscr {O}}\)-implications induced from \(C_L\)-overlap functions on complete lattices. Soft Comput 26, 8229–8243 (2022). https://doi.org/10.1007/s00500-022-07241-2

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