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On the Strongly Hopfian Acts over Semigroups
National Research University MIET, Lomonosov Moscow State University, Moscow, Russia
National Research University MIET, Lomonosov Moscow State University, Moscow, Russia
Russian Presidental Academy of National Economy and Public Administration, Moscow, Russia
- 1 National Research University MIET, Lomonosov Moscow State University, Moscow, Russia
- 2 National Research University MIET, Lomonosov Moscow State University, Moscow, Russia
- 3 Russian Presidental Academy of National Economy and Public Administration, Moscow, Russia
Applied Mathematics·Volume 16 (2025)·Pages 183–189·Published 14 February 2025·DOI10.4236/am.2025.162008
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Abstract
We find the necessary and sufficient conditions on a coproduct of connected acts over a semigroup to be strongly hopfian. From this, we deduce the conditions of the strong hopfness for unitary acts over groups. Moreover, we prove that a finite coproduct of strongly hopfian acts over an arbitrary semigroup is strongly hopfian.
KeywordsAct Over SemigroupAct Over GroupStrongly Hopfian Act
- Kurosh, A.G. (2003) The Theory of Groups, V. 1: Chelsea Publ. Co., N.Y., 1956, 1960, 272 p.; V. 2. AMS, 308 p.
- Varadarajan, K. (1992) Hopfian and Co-Hopfian Objects. Publicacions Matemà tiques , 36, 293-317. https://doi.org/10.5565/publmat_36192_21
- Deo, S. and Varadarajan, K. (1997) Hopfian and Co-Hopfian Groups. Bulletin of the Australian Mathematical Society , 56, 17-24. https://doi.org/10.1017/s0004972700030690
- Hirshon, R. (1970) On Hopfian Groups. Pacific Journal of Mathematics , 32, 753-766. https://doi.org/10.2140/pjm.1970.32.753
- Kaygorodov, E.V. (2014) Hopfian Abelian Groups. Gorno-Altaisk State Univ., 128 p. (In Russian)
- Kilp, M., Knauer, U. and Mikhalev, A.V. (2000) Monoids, Acts and Categories. W. de Gruyter, xvii + 529 p.
- Kozhukhov, I.B. and Mikhalev, A.V. (2023) Acts over Semigroups. Journal of Mathematical Sciences , 269, 362-401. https://doi.org/10.1007/s10958-023-06287-3
- Kozhukhov, I.B. and Kolesnikova, K.A. (2023) On Hopfianity and Co-Hopfianity of Acts over Groups. Journal of Mathematical Sciences , 269, 356-361. https://doi.org/10.1007/s10958-023-06286-4
- Kozhukhov, I.B. and Kolesnikova, K.A. (2023) Coproduct of Co-Hopfian Acts, Proc. VIII Int. Conf. “Donetsk Readings 2023”. Vol. 1, Donetsk State Univ., 167-169. (In Russian)
- Roueentan, M. and Khosravi, R. (2022) On Hopfian (Co-Hopfian) and Fitting S-Acts (I). 13 p.
- Roueentan, M. and Khosravi, R. (2024) A Note on Hopfian and Co-Hopfian S-Acts. Iranian Journal of Science . https://doi.org/10.1007/s40995-024-01753-2
- Kartashov, V.K. (2008) Independent Systems of Generators and the Hopf Property for Unary Algebras. Discrete Mathematics and Applications , 18, 625-630. https://doi.org/10.1515/dma.2008.047
- Clifford, A.H. and Preston, G.B. (1961, 1967) The Algebraic Theory of Semigroups, V. I, II, Mathematical Surveys, Number 7. Amer. Math. Soc., Providence, xvi + 244 p. and xv + 350 p.
- Cohn, P.M. (1965) Universal Algebra. Harper & Row, xv + 333 p.
- Jakubíková-Studenovská, D. and Pócs, J. (2009) Monounary Algebras. UPJS, 302 p.