Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

SEMIGROUPS OF TRANSFORMATIONS WITH INVARIANT SET

  • Received : 2009.09.23
  • Published : 2011.03.01
Download PDF
⟨ Previous Next ⟩

Abstract

Let T(X) denote the semigroup (under composition) of transformations from X into itself. For a fixed nonempty subset Y of X, let S(X, Y) = {${\alpha}\;{\in}\;T(X)\;:\;Y\;{\alpha}\;{\subseteq}\;Y$}. Then S(X, Y) is a semigroup of total transformations of X which leave a subset Y of X invariant. In this paper, we characterize when S(X, Y) is isomorphic to T(Z) for some set Z and prove that every semigroup A can be embedded in S($A^1$, A). Then we describe Green's relations for S(X, Y) and apply these results to obtain its group H-classes and ideals.

Keywords

References

  1. A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vol. I and II, Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I. 1961 and 1967.
  2. C. G. Doss, Certain equivalence relations in transformation semigroups, M. A. Thesis, directed by D. D. Miller, University of Tennessee, 1955.
  3. J. M. Howie, Fundamentals of Semigroup Theory, The Clarendon Press, Oxford University Press, New York, 1995.
  4. K. D. Magill Jr., Subsemigroups of S(X), Math. Japon. 11 (1966), 109-115.
  5. A. I. Malcev, Symmetric groupoids, Mat. Sbornik N. S. 31(73) (1952), 136-151.
  6. S. Nenthein, P. Youngkhong, and Y. Kemprasit, Regular elements of some transformation semigroups, Pure Math. Appl. 16 (2005), no. 3, 307-314.
  7. J. S. V. Symons, Some results concerning a transformation semigroup, J. Austral. Math. Soc. 19 (1975), no. 4, 413-425. https://doi.org/10.1017/S1446788700034455

Cited by

  1. On certain order-preserving transformation semigroups vol.45, pp.7, 2017, https://doi.org/10.1080/00927872.2016.1233333
  2. Semigroups of transformations with fixed sets vol.36, pp.1, 2013, https://doi.org/10.2989/16073606.2013.779958
  3. REGULARITY OF TRANSFORMATION SEMIGROUPS DEFINED BY A PARTITION vol.31, pp.2, 2016, https://doi.org/10.4134/CKMS.2016.31.2.217
  4. NATURAL PARTIAL ORDER IN SEMIGROUPS OF TRANSFORMATIONS WITH INVARIANT SET vol.87, pp.01, 2013, https://doi.org/10.1017/S0004972712000287
  5. Magnifying elements of semigroups of transformations with invariant set pp.1793-7183, 2019, https://doi.org/10.1142/S1793557119500566