Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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COREGULARITY OF ORDER-PRESERVING SELF-MAPPING SEMIGROUPS OF FENCES

  • JENDANA, KETSARIN (Department of Mathematics Faculty of Science Silpakorn University) ;
  • SRITHUS, RATANA (Department of Mathematics Faculty of Science Silpakorn University)
  • Received : 2015.03.10
  • Published : 2015.10.31
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Abstract

A fence is an ordered set that the order forms a path with alternating orientation. Let F = (F;${\leq}$) be a fence and let OT(F) be the semigroup of all order-preserving self-mappings of F. We prove that OT(F) is coregular if and only if ${\mid}F{\mid}{\leq}2$. We characterize all coregular elements in OT(F) when F is finite. For any subfence S of F, we show that the set COTS(F) of all order-preserving self-mappings in OT(F) having S as their range forms a coregular subsemigroup of OT(F). Under some conditions, we show that a union of COTS(F)'s forms a coregular subsemigroup of OT(F).

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References

  1. G. Bijev and K. Todorov, Coregular semigroups, Notes on Semigroups VI, 1-11, Budapest, 1980-1984.
  2. F. A. Cezus, K. D. Magill, and S. Subbiah, Maximal ideals of semigroups of endomorphsims, Bull. Aust. Math. Soc. 12 (1975), 211-225. https://doi.org/10.1017/S0004972700023820
  3. J. Chvalina and K. Matouskova, Coregularity of endomorphism monoids of unars, Arch. Math. 20 (1984), no. 1, 43-48.
  4. J. Demetrovics and L. Ronyai, Algebraic properties of crowns and fences, Order 6 (1989), no. 1, 91-99. https://doi.org/10.1007/BF00341639
  5. I. Dimitrova and J. Koppitz, Coregular semigroups of full transformations, Demonstr. Math. 44 (2011), no. 4, 739-753.
  6. J. D. Farley, The number of order-preserving maps between fences and crowns, Order 12 (1995), no. 1, 5-44. https://doi.org/10.1007/BF01108588
  7. L. M. Gluskin, Isotone transformation semigroups, Uspekhi Mat. Nauk 16 (1961), no. 5, 157-162.
  8. P. M. Higgins, D. Mitchell, and N. Ruskuc, Generating the full transformation semigroups using order-preserving mappings, Glasg. Math. J. 45 (2003), no. 3, 557-566. https://doi.org/10.1017/S0017089503001460
  9. V. I. Kim and I. B. Kozhukhov, Regularity conditions for semigroups of isotone transformations of countable chains, J. Math. Sci. 152 (2008), no. 2, 203-208. https://doi.org/10.1007/s10958-008-9063-x
  10. E. S. Lyapin, An abstract characteristic of a class of endomorphism semigroup of systems of general view, Mat. Sb. 70 (112) (1966), no. 2, 173-179.
  11. M. Ma, T. You, S. Luo, Y. Yang, and L. Wang, Regularity and Green's relation for finite E-order-preserving transformations semigroups, Semigroup Forum 80 (2010), no. 1, 164-173. https://doi.org/10.1007/s00233-009-9192-7
  12. K. D. Magill, Jr. and S. Subbiah, Green's relations for regular elements of semigroups of endomorphisms, Canad. J. Math. 26 (1974), no. 6, 1484-1497. https://doi.org/10.4153/CJM-1974-144-x
  13. I. V. Pozdnyakova, Semigroups of endomorphisms of some infinite monounary algebras, J. Math. Sci. 190 (2013), no. 5, 658-668. https://doi.org/10.1007/s10958-013-1278-9
  14. S. Radeleczki, The automorphism group of unary algebras, Math. Pannon. 7 (1996), no. 2, 253-271.
  15. A. Rutkowski, The formula for the number of order-preserving self-mappings of a fence, Order 9 (1992), no. 2, 127-137. https://doi.org/10.1007/BF00814405
  16. P. Zhao and M. Yang, Regularity and Green's relations on semigroups of transformation preserving order and compression, Bull. Korean Math. Soc. 49 (2012), no. 5, 1015-1025. https://doi.org/10.4134/BKMS.2012.49.5.1015

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