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DOI QR Code

INJECTIVE PARTIAL TRANSFORMATIONS WITH INFINITE DEFECTS

  • Singha, Boorapa ;
  • Sanwong, Jintana ;
  • Sullivan, Robert Patrick
  • Received : 2010.08.11
  • Published : 2012.01.31

Abstract

In 2003, Marques-Smith and Sullivan described the join ${\Omega}$ of the 'natural order' $\leq$ and the 'containment order' $\subseteq$ on P(X), the semigroup under composition of all partial transformations of a set X. And, in 2004, Pinto and Sullivan described all automorphisms of PS(q), the partial Baer-Levi semigroup consisting of all injective ${\alpha}{\in}P(X)$ such that ${\mid}X{\backslash}X{\alpha}\mid=q$, where $N_0{\leq}q{\leq}{\mid}X{\mid}$. In this paper, we describe the group of automorphisms of R(q), the largest regular subsemigroup of PS(q). In 2010, we studied some properties of $\leq$ and $\subseteq$ on PS(q). Here, we characterize the meet and join under those orders for elements of R(q) and PS(q). In addition, since $\leq$ does not equal ${\Omega}$ on I(X), the symmetric inverse semigroup on X, we formulate an algebraic version of ${\Omega}$ on arbitrary inverse semigroups and discuss some of its properties in an algebraic setting.

Keywords

partial transformation semigroup;Baer-Levi semigroup;inverse semigroup;natural order;containment order;meet and join

References

  1. S. Y. Chen and S. C. Hsieh, Factorizable inverse semigroups, Semigroup Forum 8 (1974), no. 4, 283-297. https://doi.org/10.1007/BF02194773
  2. A. H. Clifford and G. B. Preston, The Algebraic Theory of Semigroups, Mathematical Surveys, No. 7, vol 1 and 2, American Mathematical Society, Providence, RI, 1961 and 1967.
  3. J. M. Howie, An Introduction to Semigroup Theory, Academic Press, London, 1976.
  4. J. M. Howie, A congruence-free inverse semigroup associated with a pair of in nite cardinals, J. Aust. Math. Soc. 31 (1981), no. 3, 337-342. https://doi.org/10.1017/S1446788700019479
  5. G. Kowol and H. Mitsch, Naturally ordered transformation semigroups, Monatsh. Math. 102 (1986), no. 2, 115-138. https://doi.org/10.1007/BF01490204
  6. I. Levi, Automorphisms of normal partial transformation semigroups, Glasg. Math. J. 29 (1987), no. 2, 149-157. https://doi.org/10.1017/S0017089500006790
  7. M. P. O. Marques-Smith and R. P. Sullivan, Partial orders on transformation semigroups, Monatsh. Math. 140 (2003), no. 2, 103-118. https://doi.org/10.1007/s00605-002-0546-4
  8. H. Mitsch, A natural partial order for semigroups, Proc. Amer. Math. Soc. 97 (1986), no. 3, 384-388 https://doi.org/10.1090/S0002-9939-1986-0840614-0
  9. F. A. Pinto and R. P. Sullivan, Baer-Levi semigroups of partial transformations, Bull. Aust. Math. Soc. 69 (2004), no. 1, 87-106. https://doi.org/10.1017/S0004972700034286
  10. J. Sanwong and R. P. Sullivan, Injective transformations with equal gap and defect, Bull. Aust. Math. Soc. 79 (2009), no. 2, 327-336. https://doi.org/10.1017/S0004972708001330
  11. B. Singha, J. Sanwong, and R. P. Sullivan, Partial orders on partial Baer-Levi semigroups, Bull. Aust. Math. Soc. 81 (2010), no. 2, 195-207. https://doi.org/10.1017/S0004972709001038
  12. R. P. Sullivan, Automorphisms of transformation semigroups, J. Aust. Math. Soc. 20 (1975), 77-84. https://doi.org/10.1017/S144678870002396X
  13. R. P. Sullivan, Semigroups generated by nilpotent transformations, J. Algebra 110 (1987), no. 2, 324-343. https://doi.org/10.1016/0021-8693(87)90049-4
  14. R. P. Sullivan, Partial orders on linear transformation semigroups, Proc. Roy. Soc. Edinburgh Sect. A 135 (2005), no. 2, 413-437. https://doi.org/10.1017/S0308210500003942