DOI QR코드

DOI QR Code

Quasi-Normal Relations - a New Class of Relations

  • ROMANO, DANIEL ABRAHAM (Faculty of Education Bijeljina, East Sarajevo University)
  • Received : 2014.06.19
  • Accepted : 2014.11.27
  • Published : 2015.09.23

Abstract

In this paper, concepts of quasi-normal and dually quasi-normal relations are introduced. Characterizations of these relations are obtained. In addition, particulary we show that the anti-order relation ${\nleqslant}$ ($={\leqslant}^C$) is a (dually) quasi-normal relation if and only if the partially ordered set (X, ${\leqslant}$) is an anti-chain.

Keywords

relations;quasi-normal relations;dually quasi-normal relations

References

  1. H. J. Bandelt, Regularity and complete distributivity, Semigroup Forum, 19(1980), 123-126 https://doi.org/10.1007/BF02572509
  2. H. J. Bandelt, On regularity classes of binary relations, In : Universal Algebra and Applications, Banach Center Publications, 9(1982), 329-333
  3. D. Hardy and M. Petrich, Binary relations as lattice isomorphisms, Ann. Mat. Pura Appl, 177(1)(1999), 195-224 https://doi.org/10.1007/BF02505909
  4. Jiang Guanghao and Xu Luoshan, Conjugative Relations and Applications, Semigroup Forum, 80(1)(2010), 85-91. https://doi.org/10.1007/s00233-009-9185-6
  5. Jiang Guanghao and Xu Luoshan, Dually normal relations on sets, Semigroup Forum, 85(1)(2012), 75-80 https://doi.org/10.1007/s00233-011-9364-0
  6. Jiang Guanghao, Xu Luoshan, Cai Jin and Han Guiwen, Normal Relations on Sets and Applications, Int. J. Contemp. Math. Sciences, 6(15)(2011), 721-726
  7. G. Markowsky, Idempotents and product representations with applications to the semigroup of binary relations, Semigroup Forum, 5(1972), 95-119 https://doi.org/10.1007/BF02572880
  8. B. M. Schein, Regular elements of the semigroup of all binary relations, Semigroup Forum, 13(1976), 95-102 https://doi.org/10.1007/BF02194925
  9. K. A. Zaretskii, Regular elements of the semigroup of binary relations, Uspehi Matem. Nauk, 3(105)(1962), 177-179.
  10. K. A. Zareckii, The semigroup of binary relations, Mat. Sb., 61(1963), 291-305.
  11. Xu Xiao-quan and Liu Yingming, Relational representations of hypercontinuous lattices, In : Domain Theory, Logic, and Computation, Kluwer Academic Publisher, 2003, 65-74.