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Inner automorphisms of Clifford monoids

Year 2024, Volume: 53 Issue: 4, 1060 - 1074, 27.08.2024
https://doi.org/10.15672/hujms.1244782

Abstract

An automorphism $\phi$ of a monoid $S$ is called inner if there exists $g$ in $U_{S}$, the group of units of $S$, such that $\phi(s)=gsg^{-1}$ for all $s $ in $S$; we call $S$ nearly complete if all of its automorphisms are inner. In this paper, first we prove several results on inner automorphisms of a general monoid and subsequently apply them to Clifford monoids. For certain subclasses of the class of Clifford monoids, we give necessary and sufficient conditions for a Clifford monoid to be nearly complete. These subclasses arise from conditions on the structure homomorphisms of the Clifford monoids: all being either bijective, surjective, injective, or image trivial.

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References

  • [1] J. Araújo and J. Konieczny, General Theorems on automorphisms of semigroups and their applications, J. Austral Math. Soc. 87, 1-17, 2009.
  • [2] J. Araújo, P. V. Bünau, J. D. Mitchell and M. Neunhöffer, Computing automorphisms of semigroups, J. Symbolic Comput. 45, 373-392, 2010.
  • [3] L. M. Gluskin, Semigroups and rings of endomorphisms of linear spaces-I, Transc. of Amer. Math. Soc. 45, 105-137, 1965.
  • [4] J. M. Howie, Fundamentals of semigroup theory, volume 12 of London Mathematical Society Monographs, New series, The Clarendon Press, Oxford University Press, New York, Oxford Science Publications, 1995.
  • [5] I. Levi, Automorphisms of normal transformation semigroups, Proc. Edinburg Math. Soc. 2, 185-205, 1985.
  • [6] I. Levi, On the inner automorphisms of finite transformation semigroups, Proc. Edinburg Math. Soc. 39, 27-30, 1996.
  • [7] I. Levi, Automorphisms of normal partial transformation semigroups, Glasgow Math. J. 29, 149-157, 1987.
  • [8] A. E. Liber, On symmetric generalised groups, Math. Sbornik N. S. 33, 531-544, 1953.
  • [9] K. D. Magill, Automorphisms of semigroup of all relations on a set, Canad. Math. Bull. 9, 73-77, 1966.
  • [10] A. I. Mal’cev, Symmetric Groupoids, Math. Sbornik N. S. 31, 146-151, 1952.
  • [11] D. J. Mir, A. H. Shah and, S. A. Ahanger, On automorphisms of monotone transformation posemigroups, Asian-Eur. J. of Math, 15, 2250032, 2022.
  • [12] T. Quinn-Gregson, Homogeneity of Inverse semigroups, Internat. J. Algebra Comput. 28, 837-875, 2018.
  • [13] T. Quinn-Gregson, $\aleph_{0}$-category of semigroups II, Semigroup Forum, 102, 809-841, 2021.
  • [14] M. Samman and J. D. P. Meldrum, On Endomorphisms of Semilattices of Groups, Algebra Colloq. 12, 93-100, 2005.
  • [15] A. H. Shah, and D. J. Mir, On Automorphisms of Strong Semilattices of $\pi$-Groups, Int. J. Nonlinear Anal. Appl. 13, 3031-3036, 2022.
  • [16] A. H. Shah, D. J. Mir and N. M. Khan, On automorphisms of Strong Semilattices of groups, Mathematics Open 1, 2250002, 2022.
  • [17] I. Schreier, Uber Abbildungen einer absrakten Menge Auf ihre Telimengen, Fund, Math. 28, 261-264, 1936.
  • [18] J. Schreier and S. Ulam, Über die Automorphismen der Permutations-gruppe der natürlichen Zahlenfolge, Fundam. Math. 28, 258-260, 1937.
  • [19] R. P. Sullivan, Automorphisms of Transformation semigroups, J. Austral Math. Soc. 20, 77-84, 1975.
  • [20] E. G. Sutov, Homomorphisms of the semigroups of all partial transformations, Izv. Vyss Ucebn Zaved Mat. 3, 177-184, 1961.
  • [21] J. S. V. Symons, Normal transformation semigroups, J. Austral Math. Soc.(A), 22, 385-390, 1976.
  • [22] M. L. Vitanza, Mappings of Semigroups Associated with Ordered Pairs, Amer. Math. Monthly 73, 1078-1082, 1966.
Year 2024, Volume: 53 Issue: 4, 1060 - 1074, 27.08.2024
https://doi.org/10.15672/hujms.1244782

Abstract

Project Number

NA

References

  • [1] J. Araújo and J. Konieczny, General Theorems on automorphisms of semigroups and their applications, J. Austral Math. Soc. 87, 1-17, 2009.
  • [2] J. Araújo, P. V. Bünau, J. D. Mitchell and M. Neunhöffer, Computing automorphisms of semigroups, J. Symbolic Comput. 45, 373-392, 2010.
  • [3] L. M. Gluskin, Semigroups and rings of endomorphisms of linear spaces-I, Transc. of Amer. Math. Soc. 45, 105-137, 1965.
  • [4] J. M. Howie, Fundamentals of semigroup theory, volume 12 of London Mathematical Society Monographs, New series, The Clarendon Press, Oxford University Press, New York, Oxford Science Publications, 1995.
  • [5] I. Levi, Automorphisms of normal transformation semigroups, Proc. Edinburg Math. Soc. 2, 185-205, 1985.
  • [6] I. Levi, On the inner automorphisms of finite transformation semigroups, Proc. Edinburg Math. Soc. 39, 27-30, 1996.
  • [7] I. Levi, Automorphisms of normal partial transformation semigroups, Glasgow Math. J. 29, 149-157, 1987.
  • [8] A. E. Liber, On symmetric generalised groups, Math. Sbornik N. S. 33, 531-544, 1953.
  • [9] K. D. Magill, Automorphisms of semigroup of all relations on a set, Canad. Math. Bull. 9, 73-77, 1966.
  • [10] A. I. Mal’cev, Symmetric Groupoids, Math. Sbornik N. S. 31, 146-151, 1952.
  • [11] D. J. Mir, A. H. Shah and, S. A. Ahanger, On automorphisms of monotone transformation posemigroups, Asian-Eur. J. of Math, 15, 2250032, 2022.
  • [12] T. Quinn-Gregson, Homogeneity of Inverse semigroups, Internat. J. Algebra Comput. 28, 837-875, 2018.
  • [13] T. Quinn-Gregson, $\aleph_{0}$-category of semigroups II, Semigroup Forum, 102, 809-841, 2021.
  • [14] M. Samman and J. D. P. Meldrum, On Endomorphisms of Semilattices of Groups, Algebra Colloq. 12, 93-100, 2005.
  • [15] A. H. Shah, and D. J. Mir, On Automorphisms of Strong Semilattices of $\pi$-Groups, Int. J. Nonlinear Anal. Appl. 13, 3031-3036, 2022.
  • [16] A. H. Shah, D. J. Mir and N. M. Khan, On automorphisms of Strong Semilattices of groups, Mathematics Open 1, 2250002, 2022.
  • [17] I. Schreier, Uber Abbildungen einer absrakten Menge Auf ihre Telimengen, Fund, Math. 28, 261-264, 1936.
  • [18] J. Schreier and S. Ulam, Über die Automorphismen der Permutations-gruppe der natürlichen Zahlenfolge, Fundam. Math. 28, 258-260, 1937.
  • [19] R. P. Sullivan, Automorphisms of Transformation semigroups, J. Austral Math. Soc. 20, 77-84, 1975.
  • [20] E. G. Sutov, Homomorphisms of the semigroups of all partial transformations, Izv. Vyss Ucebn Zaved Mat. 3, 177-184, 1961.
  • [21] J. S. V. Symons, Normal transformation semigroups, J. Austral Math. Soc.(A), 22, 385-390, 1976.
  • [22] M. L. Vitanza, Mappings of Semigroups Associated with Ordered Pairs, Amer. Math. Monthly 73, 1078-1082, 1966.
There are 22 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Aftab Shah 0000-0003-1143-0199

Dilawar Mir This is me 0000-0002-1749-6968

Thomas Quinn-gregson 0000-0003-2456-3759

Project Number NA
Early Pub Date April 14, 2024
Publication Date August 27, 2024
Published in Issue Year 2024 Volume: 53 Issue: 4

Cite

APA Shah, A., Mir, D., & Quinn-gregson, T. (2024). Inner automorphisms of Clifford monoids. Hacettepe Journal of Mathematics and Statistics, 53(4), 1060-1074. https://doi.org/10.15672/hujms.1244782
AMA Shah A, Mir D, Quinn-gregson T. Inner automorphisms of Clifford monoids. Hacettepe Journal of Mathematics and Statistics. August 2024;53(4):1060-1074. doi:10.15672/hujms.1244782
Chicago Shah, Aftab, Dilawar Mir, and Thomas Quinn-gregson. “Inner Automorphisms of Clifford Monoids”. Hacettepe Journal of Mathematics and Statistics 53, no. 4 (August 2024): 1060-74. https://doi.org/10.15672/hujms.1244782.
EndNote Shah A, Mir D, Quinn-gregson T (August 1, 2024) Inner automorphisms of Clifford monoids. Hacettepe Journal of Mathematics and Statistics 53 4 1060–1074.
IEEE A. Shah, D. Mir, and T. Quinn-gregson, “Inner automorphisms of Clifford monoids”, Hacettepe Journal of Mathematics and Statistics, vol. 53, no. 4, pp. 1060–1074, 2024, doi: 10.15672/hujms.1244782.
ISNAD Shah, Aftab et al. “Inner Automorphisms of Clifford Monoids”. Hacettepe Journal of Mathematics and Statistics 53/4 (August 2024), 1060-1074. https://doi.org/10.15672/hujms.1244782.
JAMA Shah A, Mir D, Quinn-gregson T. Inner automorphisms of Clifford monoids. Hacettepe Journal of Mathematics and Statistics. 2024;53:1060–1074.
MLA Shah, Aftab et al. “Inner Automorphisms of Clifford Monoids”. Hacettepe Journal of Mathematics and Statistics, vol. 53, no. 4, 2024, pp. 1060-74, doi:10.15672/hujms.1244782.
Vancouver Shah A, Mir D, Quinn-gregson T. Inner automorphisms of Clifford monoids. Hacettepe Journal of Mathematics and Statistics. 2024;53(4):1060-74.