r-notion of Conjugacy in Partial and Full Injective Transformations

Document Type : Research Paper


Department of Mathematics, Central University of Kashmir, Ganderbal, 191201.


In this paper, we define a new notion of conjugacy in semigroups that reduces to the n-notion of conjugacy in an inverse semigroup. We compare our new notion with the existing notions. We characterize the notion in partial injective and in full injective transformations and determine the conjugacy classes in these semigroups.


[1] J. Araujo, M. Kinyon and J. Konieczny, Conjugacy in inverse semigroups, J. Algebra, 533 (2019) 142-173
[2] J. Araujo, M. Kinyon, J. Konieczny and A. Malheiro, Four notions of conjugacy for abstract semigroups, Proc. Roy. Soc. Edinburgh Sect. A: Mathematics, 147 No. 6 (2017) 1169-1214.
[3] J. Araujo, J. Konieczny and A. Malheiro, Conjugation in semigroups, J. Algebra, 403 (2014) 93-134.
[4] J. Koneiczny, A new definition of conjugacy for semigroups, J. Algebra and Appl., 17 No. 02 (2018) 1850032.
[5] G. Kudryavtseva and V. Mazorchuk, On conjugation in some transformation and Brauer-type semigroups, Publ. Math. Debrecen, 70 (2007) 19-43.
[6] G. Kudryavtseva and V. Mazorchuk, On three approaches to conjugacy in semigroups, Semigr. Forum, 78 (2009) 14-20.
[7] J. M. Howie, Fundamentals of Semigroup Theory, Oxford University Press, New York, 1995.
[8] T. Jech, Set Theory, Third Edition, Springer-Verlag, New York, 2006.
[9] G. Lallement, Semigroups and Combinatorial Applications, John Wiley and Sons, New York, 1979.
[10] F. Otto, Conjugacy in monoids with a special Church-Rosser presentation is decidable, Semigr. Forum, 29 (1984) 223-240.
[11] A. H. Shah, M. R. Parray, ~r notion of conjugacy in partial transformation semigroups, Korean J. Math., 30 No. 1 (2022) 115-125.
[12] L. Zhang, Conjugacy in special monoids, J. Algebra, 143 (1991) 487-497.
[13] L. Zhang, On the conjugacy problem for one-relator monoids with elements of finite order, Internat. J. Algebra Comput., 2 (1992) 209-220.