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Rees semigroups of digraphs for classification of data

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Abstract

Recent research has motivated the investigation of the weights of ideals in semiring constructions based on semigroups. The present paper introduces Rees semigroups of directed graphs. This new construction is a common generalization of Rees matrix semigroups and incidence semigroups of digraphs. For each finite subsemigroup \(S\) of the Rees semigroup of a digraph and for every zero-divisor-free idempotent semiring \(F\) with identity element, our main theorem describes all ideals \(J\) in the semigroup semiring \(F_0[S]\) such that \(J\) has the largest possible weight.

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References

  1. Abawajy, J., Kelarev, A.V.: A multi-tier ensemble construction of classifiers for phishing email detection and filtering, Cyberspace Safety and Security, CSS 2012. Lect. Notes Comput. Sci. 7672, 48–56 (2012)

    Article  Google Scholar 

  2. Abawajy, J., Kelarev, A.V.: Classification systems based on combinatorial semigroups. Semigroup Forum 86, 603–612 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  3. Abawajy, J.H., Kelarev, A.V., Chowdhury, M.: Multistage approach for clustering and classification of ECG data. Comput. Methods Programs in Biomed. 112, 720–730 (2013)

    Article  Google Scholar 

  4. Abawajy, J., Kelarev, A., Chowdhury, M.: Large iterative multitier ensemble classifiers for security of big data. IEEE Trans. Emerg. Top. Comput. 2, 352–363 (2014)

    Article  Google Scholar 

  5. Abawajy, J., Kelarev, A.V., Chowdhury, M., Jelinek, H.F.: Enhancing predictive accuracy of cardiac autonomic neuropathy using blood biochemistry features and iterative multi-tier ensembles, IEEE J. Biomed. Health Inform., to appear

  6. Abawajy, J., Kelarev, A., Chowdhury, M., Stranieri, A., Jelinek, H.F.: Predicting cardiac autonomic neuropathy category for diabetic data with missing values. Comput. Biol. Med. 43, 1328–1333 (2013)

    Article  Google Scholar 

  7. Abawajy, J., Kelarev, A.V., Miller, M., Ryan, J.: Incidence semirings of graphs and visible bases. Bull. Aust. Math. Soc. 89, 451–459 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  8. Abawajy, J., Kelarev, A.V., Zeleznikow, J.: Optimization of classification and clustering systems based on Munn semirings. Semigroup Forum 87, 617–626 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  9. Abawajy, J., Kelarev, A.V., Yearwood, J.L., Turville, C.: A data mining application of the incidence semirings. Houston J. Math. 39, 1083–1093 (2013)

    MATH  MathSciNet  Google Scholar 

  10. Beliakov, G., Yearwood, J., Kelarev, A.: Application of rank correlation, clustering and classification in information security. J. Netw. 7, 935–945 (2012)

    Google Scholar 

  11. Dazeley, R., Yearwood, J.L., Kang, B.H., Kelarev, A.V.: Consensus clustering and supervised classification for profiling phishing emails in internet commerce security. Lect. Notes in Comput. Sci. 6232, 235–246 (2010)

    Article  Google Scholar 

  12. Gao, D.Y., Kelarev, A.V., Yearwood, J.L.: Optimization of matrix semirings for classification systems. Bull. Aust. Math. Soc. 84, 492–503 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  13. Golan, J.S.: Semirings and Their Applications. Kluwer Academic Publishers, Dordrecht (1999)

    Book  MATH  Google Scholar 

  14. Hall, T.E., Kublanovskii, S.I., Margolis, S., Sapir, M.V., Trotter, P.G.: Algorithmic problems for finite groups and finite 0-simple semigroups. J. Pure Appl. Algebra 119, 75–96 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  15. Howie, J.M.: Fundamentals of Semigroup Theory. Clarendon Press, Oxford (1995)

    MATH  Google Scholar 

  16. Jackson, M., Volkov, M.: Undecidable problems for completely 0-simple semigroups. J. Pure Appl. Algebra 213, 1961–1978 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  17. Jelinek, H.F., Abawajy, J.H., Kelarev, A.V., Chowdhury, M.U., Stranieri, A.: Decision trees and multi-level ensemble classifiers for neurological diagnostics. AIMS Med. Sci. 1(1), 1–12 (2014)

    Article  Google Scholar 

  18. Kelarev, A.V.: Ring Constructions and Applications. World Scientific, River Edge (2002)

    MATH  Google Scholar 

  19. Kelarev, A.V.: Graph Algebras and Automata. Marcel Dekker, New York (2003)

    MATH  Google Scholar 

  20. Kelarev, A.V., Abawajy, J., Stranieri, A., Jelinek, H.F.: Empirical investigation of decision tree ensembles for monitoring cardiac complications of diabetes. Int. J. Data Warehous. Min. 9, 1–18 (2013)

    Article  Google Scholar 

  21. Kelarev, A., Brown, S., Watters, P., Wu, X.-W., Dazeley, R.: Establishing reasoning communities of security experts for internet commerce security, In: “Technologies for Supporting Reasoning Communities and Collaborative Decision Making: Cooperative Approaches”, IGI Global, 2011, pp. 380–396

  22. Kelarev, A.V., Passman, D.S.: A description of incidence rings of group automata. Contemp. Math. 456, 27–33 (2008)

    Article  MathSciNet  Google Scholar 

  23. Kelarev, A., Ryan, J., Yearwood, J.: Cayley graphs as classifiers for data mining: The influence of asymmetries. Discrete Math. 309, 5360–5369 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  24. Kelarev, A., Ryan, J., Yearwood, J.: An algorithm for the optimization of multiple classifiers in data mining based on graphs. J. Comb. Math. Comb. Comput. 71, 65–86 (2009)

    MATH  MathSciNet  Google Scholar 

  25. Kelarev, A.V., Watters, P.W., Yearwood, J.L.: Rees matrix constructions for clustering of data. J. Aust. Math. Soc. 87, 377–393 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  26. Kelarev, A.V., Yearwood, J.L., Mammadov, M.A.: A formula for multiple classifiers in data mining based on Brandt semigroups. Semigroup Forum 78, 293–309 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  27. Kelarev, A.V., Yearwood, J.L., Vamplew, P.W.: A polynomial ring construction for classification of data. Bull. Aust. Math. Soc. 79, 213–225 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  28. Kelarev, A.V., Yearwood, J.L., Vamplew, P.W., Abawajy, J., Chowdhury, M.: Optimization and matrix constructions for classification of data. NZ. J. Math. 41, 65–73 (2011)

    MATH  MathSciNet  Google Scholar 

  29. Kelarev, A.V., Yearwood, J.L., Watters, P.A.: Rees matrix constructions for clustering of data. J. Aust. Math. Soc. 87, 377–393 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  30. Kelarev, A.V., Yearwood, J.L., Watters, P.A.: Optimization of classifiers for data mining based on combinatorial semigroups. Semigroup Forum 82, 242–251 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  31. Kelarev, A.V., Yearwood, J.L., Watters, P., Wu, X.W., Abawajy, J.H., Pan, L.: Internet security applications of the Munn rings. Semigroup Forum 81, 162–171 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  32. Kelarev, A.V., Yearwood, J.L., Zi, Lifang: Optimal Rees matrix constructions for analysis of data. J. Aust. Math. Soc. 92, 357–366 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  33. Pastijn, F.: Weak commutativity in idempotent semirings. Semigroup Forum 72, 283–311 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  34. Reilly, N.R.: Bipartite graphs and completely 0-simple semigroups. Semigroup Forum 84, 176–199 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  35. Spiegel, E., O’Donnell, C.J.: Incidence Algebras. Marcel Dekker, New York (1997)

    MATH  Google Scholar 

  36. Stranieri, A., Abawajy, J., Kelarev, A.V., Huda, S., Chowdhury, M., Jelinek, H.F.: An approach for Ewing test selection to support the clinical assessment of cardiac autonomic neuropathy. Artif. Intell. Med. 58, 185–193 (2013)

    Article  Google Scholar 

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Acknowledgments

The first author was supported by Discovery grant DP0880501 from Australian Research Council. The second author was supported by ARC Discovery Grant DP0449469. The third author was supported by Discovery grant DP0450294.

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Authors and Affiliations

  1. School of Information Technology, Deakin University, 221 Burwood, Melbourne, VIC, 3125, Australia

    J. Abawajy & A. V. Kelarev

  2. CARMA Priority Research Centre, School of Mathematical and Physical Sciences, University of Newcastle, University Drive, Callaghan, NSW, 2308, Australia

    M. Miller

  3. Department of Mathematics, University of West Bohemia, Pilsen, Czech Republic

    M. Miller

  4. School of Electrical Engineering and Computer Science, University of Newcastle, University Drive, Callaghan, NSW, 2308, Australia

    J. Ryan

Authors
  1. J. Abawajy
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  2. A. V. Kelarev
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  3. M. Miller
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  4. J. Ryan
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Correspondence to A. V. Kelarev.

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Communicated by Marcel Jackson.

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Abawajy, J., Kelarev, A.V., Miller, M. et al. Rees semigroups of digraphs for classification of data. Semigroup Forum 92, 121–134 (2016). https://doi.org/10.1007/s00233-014-9685-x

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  • DOI: https://doi.org/10.1007/s00233-014-9685-x

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