Skip to main content
SpringerLink
Log in

Homomorphism between ordered decision systems

  • Foundations
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

Communication between information systems is an important topic in granular computing. The notion of homomorphism is viewed as a basic tool to study this kind of problems. This work studies basic properties of ordered decision systems under homomorphism. We first review consistent function related to ordered relation and introduce the notion of consistent function related to a universal subset. The relationship between the two kinds of consistent functions is given. Then, the relationships between ordered rough approximations and their images are discussed under consistent functions. Finally, ordered decision systems are divided into two classes: consistent and inconsistent ordered decision systems. For each type of decision system, some of its basic homomorphic properties are presented. It is proved that attribute reductions in an original system and its image system are equivalent to each other under the condition of homomorphism in each type of ordered decision system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Bryson N, Mobolurin A (1996) An action learning evaluation procedure for multiple criteria decision making problems. Eur J Oper Res 96:379–386

    Article  MATH  Google Scholar 

  • Chen Y (2016) An adjustable multigranulation fuzzy rough set. Int J Mach Learn Cybern 7(2):267–274

    Article  Google Scholar 

  • Dai J, Xu Q (2013) Attribute selection based on information gain ratio in fuzzy rough set theory with application to tumor classification. Appl Soft Comput 13(1):211–221

    Article  Google Scholar 

  • Dai J, Hu H, Wu W, Qian Y, Huang D (2017) Maximal discernibility pairs based approach to attribute reduction in fuzzy rough sets. IEEE Trans Fuzzy Syst. https://doi.org/10.1109/TFUZZ.2017.2768044

    Google Scholar 

  • Deng T, Chen Y, Xu W, Dai Q (2007) A novel approach to fuzzy rough sets based on a fuzzy covering. Inf Sci 177:2308–2326

    Article  MathSciNet  MATH  Google Scholar 

  • Dick S, Schenker A, Pedrycz W, Kandel A (2007) Regranulation: a granular algorithm enabling communication between granular worlds. Inf Sci 177:408–435

    Article  MathSciNet  MATH  Google Scholar 

  • Gong Z, Xiao Z (2010) Communicating between information systems based on including degrees. Int J Gen Syst 39(2):189–206

    Article  MathSciNet  MATH  Google Scholar 

  • Greco S, Matarazzo B, Slowinski R (2001) Rough sets theory for multicriteria decision analysis. Eur J Oper Res 129:11–47

    Article  MATH  Google Scholar 

  • Greco S, Matarazzo B, Slowinski R (2002) Rough sets methodology for sorting problems in presence of multiple attributes and criteria. Eur J Oper Res 138:247–259

    Article  MathSciNet  MATH  Google Scholar 

  • Grzymala-Busse JW (1986) Algebraic properties of knowledge representation systems. In: Proceedings of the ACM SIGART international symposium on methodologies for intelligent systems, Knoxville, pp 432–440

  • Grzymala-Busse JW, Sedelow WA Jr (1998) On rough sets, and information system homomorphism. Bull Pol Acad Sci Tech Sci 36(3&4):233–239

    MATH  Google Scholar 

  • Iyer NS (2003) A family of dominance rules for multiattribute decision making under uncertainty. IEEE Trans Syst Man Cybern Part A 33:441–450

    Article  Google Scholar 

  • Kryszkiewski M (2001) Comparative study of alternative type of knowledge reduction in inconsistent systems. Int J Intell Syst 16:105–120

    Article  Google Scholar 

  • Lang G, Li Q, Guo L (2015a) Homomorphisms between covering approximation spaces. Fundam Inform 138(3):351–371

    MathSciNet  MATH  Google Scholar 

  • Lang G, Li Q, Guo L (2015b) Homomorphisms-based attribute reduction of dynamic fuzzy covering information systems. Int J Gen Syst 44(7–8):791–811

    Article  MathSciNet  MATH  Google Scholar 

  • Li D, Ma Y (2000) Invariant characters of information systems under some homomorphisms. Inf Sci 129:211–220

    Article  MathSciNet  MATH  Google Scholar 

  • Li Z, Liu Y, Li Q, Qin B (2016) Relationships between knowledge bases and related results. Knowl Inf Syst 49(1):171–19

    Article  Google Scholar 

  • Mi J, Zhang W (2004) An axiomatic characterization of a fuzzy generalization of rough sets. Inf Sci 160(1–4):235–249

    Article  MathSciNet  MATH  Google Scholar 

  • Pawlak Z (1982) Rough sets. Int J Comput Inf Sci 11:341–356

    Article  MATH  Google Scholar 

  • Pedrycz W, Vukovich G (2000) Granular worlds: representation and communication problems. Int J Intell Syst 15:1015–1026

    Article  MATH  Google Scholar 

  • Pedrycz W, Bezdek JC, Hathaway RJ, Rogers GW (1998) Two nonparametric models for fusing heterogeneous fuzzy data. IEEE Trans Fuzzy Syst 6(3):411–425

    Article  Google Scholar 

  • Shao M, Zhang W (2005) Dominance relation and rules in an incomplete ordered information system. Int J Intell Syst 20:13–27

    Article  MATH  Google Scholar 

  • She Y, He X (2014) Uncertainty measures in rough algebra with applications to rough logic. Int J Mach Learn Cybern 5(5):671–681

    Article  Google Scholar 

  • Slowinski R, Vanderpooten D (2000) A generalized definition of rough approximations based on similarity. IEEE Trans Knowl Data Eng 12(2):331–336

    Article  Google Scholar 

  • Sun L, Xu J, Tian Y (2012) Feature selection using rough entropy-based uncertainty measures in incomplete decision systems. Knowl Based Syst 36:206–216

    Article  Google Scholar 

  • Sun B, Ma W, Zhao H (2016) An approach to emergency decision-making based on decision-theoretic rough set over two universes. Soft Comput 20(9):3617–3628

    Article  Google Scholar 

  • Sun B, Ma W, Qian Y (2017a) Multigranulation fuzzy rough set over two universes and its application to decision making. Knowl Based Syst 123:61–74

    Article  Google Scholar 

  • Sun L, Zhang X, Xu J, Wang W, Liu R (2018) A gene selection approach based on the fisher linear discriminant and the neighborhood rough set. Bioengineered 9(1):144–151. https://doi.org/10.1080/21655979.2017

  • Tsang ECC, Wang C, Chen D, Wu C, Hu Q (2013) Communication between information systems using fuzzy rough sets. IEEE Trans Fuzzy Syst 21(3):527–540

    Article  Google Scholar 

  • Wang C, Wu C, Wu Cong (2008) Communicating between information systems. Inf Sci 178:3228–3239

    Article  MathSciNet  MATH  Google Scholar 

  • Wang C, Chen D, Hu Q (2010) Some invariant properties of ordered information systems under homomorphism. Sci China Ser F Inf Sci 53:1816–1825

    Article  MathSciNet  Google Scholar 

  • Wang C, Chen D, Sun B, Hu Q (2012) Communication between information systems with covering rough sets. Inf Sci 216:17–33

    Article  MathSciNet  MATH  Google Scholar 

  • Wang C, Shao M, He Q, Qian Y, Qi Y (2016) Feature subset selection based on fuzzy neighborhood rough sets. Knowl Based Syst 111(1):173–179

    Article  Google Scholar 

  • Wang C, He Q, Shao M, Hu Q (2017a) Feature selection based on maximal neighborhood discernibility. Int J Mach Learn Cybern. https://doi.org/10.1007/s13042-017-0712-6

    Google Scholar 

  • Wang C, Hu Q, Wang X, Chen D, Qian Y (2017b) Feature selection based on neighborhood discrimination index. IEEE Trans Neural Netw Learn Syst. https://doi.org/10.1109/TNNLS.2017.2710422

    Google Scholar 

  • Wu W (2008) Attribute reduction based on evidence theory in incomplete decision systems. Inf Sci 178:1355–1371

    Article  MathSciNet  MATH  Google Scholar 

  • Yang X, Qi Y, Yu D, Yu H, Yang J (2015) \(\alpha \)-Dominance relation and rough sets in interval-valued information systems. Inf Sci 294:334–347

    Article  MathSciNet  MATH  Google Scholar 

  • Yang Y, Chen D, Wang H, Tsang ECC, Zhang D (2017a) Fuzzy rough set based incremental attribute reduction from dynamic data with sample arriving. Fuzzy Sets Syst 312:66–86

    Article  MathSciNet  MATH  Google Scholar 

  • Yang Y, Chen D, Wang H (2017b) Active sample selection based incremental algorithm for attribute reduction with rough sets. IEEE Trans Fuzzy Syst 25(4):825–838

    Article  Google Scholar 

  • Yao YY (2008) Probabilistic rough set approximations. Int J Approx Reason 49(2):255–271

    Article  MATH  Google Scholar 

  • Zhang H, Yang S (2016) Inclusion measure for typical hesitant fuzzy sets, the relative similarity measure and fuzzy entropy. Soft Comput 20(4):1277–1287

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang H, Yang S (2017) Feature selection and approximate reasoning of large-scale set-valued decision tables based on alpha-dominance-based quantitative rough sets. Inf Sci 378:328–347

    Article  Google Scholar 

  • Zhao S, Chen H, Li C, Zhai M (2013) RFRR: robust fuzzy rough reduction. IEEE Trans Fuzzy Syst 21(5):825–841

    Article  Google Scholar 

  • Zhao S, Chen H, Li C, Du X, Sun H (2015) A novel approach to building a robust fuzzy rough classifier. IEEE Trans Fuzzy Syst 23(4):769–786

    Article  Google Scholar 

  • Zhu W (2007) Generalized rough sets based on relations. Inf Sci 177:4997–5011

    Article  MathSciNet  MATH  Google Scholar 

  • Zhu P, Wen Q (2010) Some improved results on communication between information systems. Inf Sci 180(18):3521–3531

    Article  MathSciNet  MATH  Google Scholar 

  • Zhu P, Wen Q (2011) Homomorphisms between fuzzy information systems revisited. Appl Math Lett 24(9):1548–1553

    Article  MathSciNet  MATH  Google Scholar 

  • Zhu P, Xie H, Wen Q (2014) A unified definition of consistent functions. Fundam Inform 135(3):331–340

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grants 61572082, 61673396, and 61473111, the Foundation of Educational Committee of Liaoning Province (LZ2016003), the Natural Science Foundation of Liaoning Province (20170540012, 20170540004).

Author information

Authors and Affiliations

  1. Department of Mathematics, Bohai university, Jinzhou, 121000, People’s Republic of China

    Changzhong Wang, Yang Huang & Xiaodong Fan

  2. Key Laboratory of Digital Publishing Big Data Mining Governance and Presentation Technology Standard, Bohai University, Jinzhou, 121013, People’s Republic of China

    Changzhong Wang & Xiaodong Fan

  3. College of Computer and Communication Engineering, China University of Petroleum, Qingdao, 266580, Shandong, People’s Republic of China

    Mingwen Shao

Authors
  1. Changzhong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yang Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiaodong Fan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Mingwen Shao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Changzhong Wang.

Ethics declarations

Conflict of interest

Authors declare that they have no conflict of interest.

Human and animal rights disclosure

This article does not contain any studies with human participants or animals performed by any of the authors.

Additional information

Communicated by A. Di Nola.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, C., Huang, Y., Fan, X. et al. Homomorphism between ordered decision systems. Soft Comput 23, 365–374 (2019). https://doi.org/10.1007/s00500-018-3156-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-018-3156-3

Keywords

Search

Navigation