We determine the relative rank of the semigroup \( \mathcal{T}\left(X,Y\right) \) of all transformations on a finite chain X with restricted range Y ⊆ X modulo the set \( \mathcal{OP}\left(X,Y\right) \) of all orientation-preserving transformations in \( \mathcal{T}\left(X,Y\right). \) Moreover, we determine the relative rank of the semigroup \( \mathcal{OP}\left(X,Y\right) \) modulo the set \( \mathcal{O}\left(X,Y\right) \) of all order-preserving transformations in \( \mathcal{OP}\left(X,Y\right). \) In both cases, we characterize the minimal relative generating sets.
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References
P. M. Catarino and P. M. Higgins, “The monoid of orientation-preserving mappings on a chain,” Semigroup Forum, 58, 190–206 (1999).
I. Dimitrova, V. H. Fernandes, and J. Koppitz, “A note on generators of the endomorphism semigroup of an infinite countable chain,” J. Alg. Appl., 16, No. 2, Article 1750031 (2017).
I. Dimitrova, J. Koppitz, and K. Tinpun, “On the relative rank of the semigroup of orientation-preserving transformations with restricted range,” Proc. 47th Spring Conference of the Union of Bulgarian Mathematicians, 109–114 (2018).
V. H. Fernandes, P. Honyam, T. M. Quinteiro, and B. Singha, “On semigroups of endomorphisms of a chain with restricted range,” Semigroup Forum, 89, 77–104 (2014).
V. H. Fernandes, P. Honyam, T. M. Quinteiro, and B. Singha, “On semigroups of orientation-preserving transformations with restricted range,” Comm. Algebra, 44, 253–264 (2016).
V. H. Fernandes and J. Sanwong, “On the rank of semigroups of transformations on a finite set with restricted range,” Algebra Colloq., 21, 497–510 (2014).
G. M. S. Gomes and J. M. Howie, “On the rank of certain semigroups of order-preserving transformations,” Semigroup Forum, 51, 275–282 (1992).
G. M. S. Gomes and J. M. Howie, “On the ranks of certain finite semigroups of transformations,” Math. Proc. Cambridge Philos. Soc., 101, 395–403 (1987).
P. M. Higgins, J. D. Mitchell, and N. Ruškuc, “Generating the full transformation semigroup using order preserving mappings,” Glasgow Math. J., 45, 557–566 (2003).
J. M. Howie, Fundamentals of Semigroup Theory, Oxford Univ. Press, Oxford (1995).
J. M. Howie and R. B. McFadden, “Idempotent rank in finite full transformation semigroups,” Proc. Roy. Soc. Edinburgh, Sec. A, 114, 161–167 (1990).
J. M. Howie, N. Ruškuc, and P. M. Higgins, “On relative ranks of full transformation semigroups,” Comm. Algebra, 26, 733–748 (1998).
D. McAlister, “Semigroups generated by a group and an idempotent,” Comm. Algebra, 26, 515–547 (1998).
N. Ruškuc, “On the rank of completely 0-simple semigroups,” Math. Proc. Cambridge Philos. Soc., 116, 325–338 (1994).
J. S. V. Symons, “Some results concerning a transformation semigroup,” J. Aust. Math. Soc., 19, 413–425 (1975).
K. Tinpun and J. Koppitz, “Relative rank of the finite full transformation semigroup with restricted range,” Acta Math. Univ. Comenian. (N.S.), 85, No. 2, 347–356 (2016).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, No. 5, pp. 617–626, May, 2021. Ukrainian DOI: 10.37863/umzh.v73i5.288.
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Dimitrova, I., Koppitz, J. On Relative Ranks of Finite Transformation Semigroups with Restricted Range. Ukr Math J 73, 718–730 (2021). https://doi.org/10.1007/s11253-021-01955-6
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DOI: https://doi.org/10.1007/s11253-021-01955-6