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Topology on Semi-Well Ordered Sets

  • Angela Sunny (Department of Mathematics, University of Calicut) ;
  • P. Sini (Department of Mathematics, University of Calicut)
  • 투고 : 2023.04.02
  • 심사 : 2023.09.04
  • 발행 : 2024.03.31

초록

A semi-well ordered set is a partially ordered set in which every non-empty subset of it contains a least element or a greatest element. It is defined as an extension of the concept of well ordered sets. An attempt is made to identify the properties of a semi-well ordered set equipped with the order topology.

키워드

과제정보

The authors are grateful to late Prof. P. T. Ramachandran for interesting discussions and useful comments. The first author is thankful for the financial support provided by the University Grants Commission, Government of India. The authors are also thankful to the reviewers and editors for their valuable comments, which helped to improve the presentation of the paper.

참고문헌

  1. Alo, A., Richard and Frink, Orrin: Topologies of chains, Math. Annalen 171(1967), 239-246.  https://doi.org/10.1007/BF01362041
  2. Alo, R.A., A proof of the complete normality of chains, Acta Mathematica Academiae Scientiarum Hungaricae Tomus, 22(3-4)(1971), 393-395.  https://doi.org/10.1007/BF01896435
  3. K. Burns and B. Hasselblatt, The Sharkovsky theorem: a natural direct proof, Amer. Math. Monthly 118(3)(March 2011), 229-244.  https://doi.org/10.4169/amer.math.monthly.118.03.229
  4. de Jongh, D.H.J. and R. Parikh, Well-partial ordering and hierarchies, Indigationes Math. 39(1979), 195-207. 
  5. G. Higman, Ordering divisibility in abstract algebras, Proc London Math. Soc. (3) 2(1)(1952), 326-336.  https://doi.org/10.1112/plms/s3-2.1.326
  6. R. D. Kopperman, E. H. Kronhemer, and R.G. Wilson, Topologies on totally ordered sets, Topology Appl., 90(1998), Elsevier, 165-185.  https://doi.org/10.1016/S0166-8641(97)00191-0
  7. J. B. Kruskal, The theory of well-quasi-ordering: a frequently discovered concept, J. Combin. Theory, Ser. A, 13(1972), 297-305.  https://doi.org/10.1016/0097-3165(72)90063-5
  8. J. B. Kruskal, Well-quasi-ordering, the tree theorem, and Vazsonyi's conjecture, Trans. Amer. Math. Soc., 95(1960), 210-225. 
  9. Malicki, Maciej and Rutkowski, Aleksander, On operations and linear extensions of well partially ordered sets, order, 21(2004), 7-17.  https://doi.org/10.1007/s11083-004-2738-0
  10. E. C. Milnerand N. Sauer, On chains and antichains in well founded partially ordered sets, J. London Math. Soc. (2), 24(1981), 1533. 
  11. P.T, Ramachandran, Anti homogeneity is equivalent to hereditary rigidity, Ganita, (1), 35(1984), 1-9. 
  12. P.T, Ramachandran, Some problems in set topology relating group, homeomorphisms and order, Ph.D. Thesis, Submitted to University of Cochin, (1985), 21-28. 
  13. D. Schmidt, The relation between the height of a well-founded partial ordering and the order types of its chains and antichains, J. Combin. Theory Ser. B, 31(2)(1981), 183-189.  https://doi.org/10.1016/S0095-8956(81)80023-8
  14. M. Sreeja, On The Total Negation of Rigidity, Int. J. Pure Appl. Math., 82(3)(2013), 391-398. 
  15. L. A. Steen and J. A. Jr. Seebach, Counter examples in topology, Dover Publications, INC., New York(1978). 
  16. Sunny, Angela and P, Sini: Semi-well ordered sets and semi-ordinals, communicated. 
  17. E. S. Wolk, Partially well ordered sets and partial ordinals, Fund. Math., 60(2)(1967), 175-186.  https://doi.org/10.4064/fm-60-2-175-186
  18. E. S. Wolk, Topology on a partially well ordered set, Fund. Math., 62(1968), 255-264. https://doi.org/10.4064/fm-62-3-255-264