Abstract
The conjugacy relation and normal subgroups play an important role in group theory. These notions are linked by the fact that the normal subgroups of a group are precisely the subgroups that are closed under the group conjugacy. There have been several attempts to extend the notion of conjugacy to semigroups. For each semigroup conjugacy, we can define the normal subsemigroups of a semigroup with respect to that conjugacy. In this paper, we study the normal subsemigroups of finite transformation semigroups. We consider four notions of conjugacy for semigroups, which have already appeared in the literature, and the semigroups \(P_n\) of partial transformations, \(T_n\) of full transformations, and \({\mathcal {I}}_n\) of partial injective transformations on a finite set with n elements. We describe the normal subsemigroups of \(P_n\), \(T_n\), and \({\mathcal {I}}_n\), with respect to each of the four conjugacy relations.
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References
Araújo, J., Bentz, W., Konieczny, J.: Directed graphs of inner translations of semigroups. Semigroup Forum 94, 650–673 (2017)
Araújo, J., Kinyon, M., Konieczny, J.: Conjugacy in inverse semigroups. J. Algebra 533, 142–173 (2019)
Araújo, J., Kinyon, M., Konieczny, J., Malheiro, A.: Four notions of conjugacy for abstract semigroups. Proc. Roy. Soc. Edinb. Sect. A 147, 1169–1214 (2017)
Araújo, J., Kinyon, M., Konieczny, J., Malheiro, A.: Decidability and independence of conjugacy problems in finitely presented monoids. Theoret. Comput. Sci. 731, 88–98 (2018)
Araújo, J., Konieczny, J.: Centralizers in the full transformation semigroup. Semigroup Forum 86, 1–31 (2013)
Araújo, J., Konieczny, J., Malheiro, A.: Conjugation in semigroups. J. Algebra 403, 93–134 (2014)
Droms, S.V., Konieczny, J., Palomba, R.: \(S_n\)-normal semigroups of partial transformations. Algebra Colloq. 19, 947–970 (2012)
Evseev, A.E., Podran, N.E.: Transformation semigroups of a finite set that are generated by idempotents with given projective characteristics. Izv. Vysš. Ucebn. Zaved. Matematika 12(103), 30–36 (1970). (Russian)
Ganyushkin, O., Mazorchuk, V.: Classical Finite Transformation Semigroups: An Introduction, Algebra and Applications, vol. 9. Springer, London (2010)
Harary, F.: The number of functional digraphs. Math. Ann. 138, 203–210 (1959)
Howie, J.M., McFadden, R.B.: Idempotent rank in finite full transformation semigroups. Proc. R. Soc. Edinb. Sect. A 114, 161–167 (1990)
Konieczny, J.: A new definition of conjugacy for semigroups, J. Algebra Appl. 17, article no. 1850032 (2018)
Kudryavtseva, G., Mazorchuk, V.: On conjugation in some transformation and Brauer-type semigroups. Publ. Math. Debrecen 70, 19–43 (2007)
Kudryavtseva, G., Mazorchuk, V.: On three approaches to conjugacy in semigroups. Semigroup Forum 78, 14–20 (2009)
Lallement, G.: Semigroups and Combinatorial Applications. Wiley, New York (1979)
Levi, I.: Normal semigroups of one-to-one transformations. Proc. Edinb. Math. Soc. (2) 34, 65–76 (1991)
Levi, I.: Injective endomorphisms of \(G_X\)-normal semigroups: infinite defects. Semigroup Forum 45, 9–22 (1992)
Levi, I.: Green’s relations on \(G_X\)-normal semigroups. Math. Jpn. 39, 19–28 (1994)
Levi, I.: Injective endomorphisms of \(G_X\)-normal semigroups: finite defects. J. Austral. Math. Soc. Ser. A 57, 261–280 (1994)
Levi, I., Seif, S.: On congruences of \(G_X\)-normal semigroups. Semigroup Forum 43, 93–113 (1991)
Levi, I., Williams, W.: Normal semigroups of partial transformations I. In: Lattices. Semigroups, and Universal Slgebra (Lisbon, 1988), pp. 173–183. Plenum Press, New York (1990)
Levi, I., Williams, W.: Normal semigroups of partial one-to-one transformations II. Semigroup Forum 43, 344–356 (1991)
Lipscomb, S., Konieczny, J.: Classification of \(S_n\)-normal semigroups. Semigroup Forum 51, 73–86 (1995)
Mesyan, Z.: Monoids of injective maps closed under conjugation by permutations. Israel J. Math. 189, 287–305 (2012)
Otto, F.: Conjugacy in monoids with a special Church-Rosser presentation is decidable. Semigroup Forum 29, 223–240 (1984)
Scott, W.R.: Group Theory. Prentice-Hall, Englewood Cliffs (1964)
Symons, J.S.V.: Normal transformation semigroups. J. Austral. Math. Soc. Ser. A 22, 385–390 (1976)
Zhang, L.: Conjugacy in special monoids. J. Algebra 143, 487–497 (1991)
Zhang, L.: On the conjugacy problem for one-relator monoids with elements of finite order. Int. J. Algebra Comput. 2, 209–220 (1992)
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Communicated by Ganna Kudryavtseva.
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Konieczny, J. Normal subsemigroups of finite transformation semigroups. Semigroup Forum 107, 680–691 (2023). https://doi.org/10.1007/s00233-023-10383-w
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DOI: https://doi.org/10.1007/s00233-023-10383-w