A higher version of Zappa products for monoids

Abstract

For arbitrary monoids $A$ and $B$, a presentation for the restricted wreath product of $A$ by $B$ that is known as the semi-direct product of $A^{\oplus B}$ by $B$ has been widely studied. After that a presentation for the Zappa product of $A$ by $B$ was defined which can be thought as the mutual semidirect product of given these two monoids under a homomorphism $\psi : A \rightarrow \mathcal{T}(B)$ and an anti-homomorphism $\delta : B \rightarrow \mathcal{T}(A)$ into the full transformation monoid on $B$, respectively on $A$. As a next step of these above results, by considering the monoids $A^{\oplus B}$ and $B^{\oplus A}$, we first introduce an extended version (generalization) of the Zappa product and then we prove the existence of an implicit presentation for this new product. Furthermore we present some other outcomes of the main theories in terms of finite and infinite cases, and also in terms of groups. At the final part of this paper we point out some possible future problems related to this subject.

Keywords

• Knit products
• Wreath products
• Zappa products
• presentations

Supporting Institution

King Abdulaziz University, Deanship of Scientific Research (DSR)

Project Number

G: 1711-130-1440

Thanks

This work was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant no. G: 1711-130-1440. The authors, therefore, acknowledge with thanks DSR for technical and financial support.

References

1. [1] F. Ates and A.S. Cevik, Knit products of finite cyclic groups and their applications, Rend. Sem. Mat. Univ. Padova, 121, 1–12, 2009.
2. [2] H. Ayik, C.M. Campbell, J.J. O’Connor and N. Ruskuc, On the efficiency of wreath products of groups, Groups-Korea 98, in: Proceedings of the International Conference held at Pusan National University, Pusan, Korea, August 10-16, 1998, Walter de Gruyter, 39–51, 2000.
3. [3] H. Ayik, C.M. Campbell, J.J. O’Connor and N. Ruskuc, Minimal presentations and efficiency of semigroups, Semigroup Forum, 60, 231–242, 2000.
4. [4] H. Ayik, F. Kuyucu and B. Vatansever, On semigroup presentations and efficiency, Semigroup Forum, 65, 329–335, 2002.
5. [5] A. Ballester-Bolinches, E. Cosme-Llopez and R. Esteban-Romero, Group extensions and graphs, Expo. Math. 34 (3), 327–334, 2016.
6. [6] A. Ballester-Bolinches, R. Esteban-Romero and M. Asaad, Products of Finite Groups, de Gruyter Exp. Math. 53, Walter de Gruyter, 2010.
7. [7] A. Ballester-Bolinches, J.E. Pin and X. Soler-Escriva, Formations of finite monoids and formal languages: Eilenberg’s variety theorem revisited, Forum Math. 26 (6), 1737–1761, 2014.
8. [8] A. Ballester-Bolinches, E. Cosme-Llopez, R. Esteban-Romero and J.J.M.M. Rutten, Formations of monoids, congruences, and formal languages, Sci. Ann. Comput. Sci. 25 (2), 171–209, 2015.

9. [9] A. Ballester-Bolinches, L.M. Ezquerro, A.A. Heliel and M.M. Al-Shomrani, Some results on products of finite groups, Bull. Malays. Math. Sci. Soc. 40 (3), 1341–1351, 2017.
10. [10] G. Baumslag, Wreath products and finitely presented groups, Math. Z. 75, 22–28, 1961.
11. [11] L.A. Bokut, Unsolvability of the word problem, and subalgebras of finitely presented Lie algebras, Izv. Akad. Nauk. SSSR Ser. Math. 36, 1173–1219, 1972.
12. [12] L.A. Bokut, Y. Chen and X. Zhao, Gröbner-Shirshov bases for free inverse semigroups, Internat. J. Algebra Comput. 19 (2), 129–143, 2009.
13. [13] M.G. Brin, On the Zappa-Szép product, Comm. Algebra 33, 393–424, 2005.
14. [14] A.S. Cevik, The efficiency of standard wreath product, Proc. Edinburgh Math. Soc. 43 (2), 415–423, 2000.
15. [15] A.S. Cevik, Minimal but inefficient presentations of the semi-direct product of some monoids, Semigroup Forum, 66 (1), 1–17, 2003.
16. [16] N.D. Gilbert and S. Wazzan, Zappa-Szép products of bands and groups, Semigroup Forum, 77, 438–455, 2008.
17. [17] A.A. Heliel, A. Ballester-Bolinches, R. Esteban-Romero and M.O. Almestady, Z- permutable subgroups of finite groups, Monat. Math. 179 (4), 523–534, 2016.
18. [18] J.M. Howie and N. Ruskuc, Constructions and presentations for monoids, Comm. Algebra, 22 (15), 6209–6224, 1994.
19. [19] J.M. Howie, Fundamentals of Semigroup Theory, London Math. Soc. Monographs, Oxford University Press, 1995.
20. [20] D.L. Johnson, Presentation of Groups, London Math. Soc. Lecture Note Series 15, Cambridge University Press, 1990.
21. [21] C. Kocapinar, E.G. Karpuz, F. Ates and A.S. Cevik, Gröbner-Shirshov bases of the generalized Bruck-Reilly -extension, Algebra Colloq. 19, 813–820, 2012.
22. [22] M. Kunze, Zappa products, Acta Math. Hung. 41, 225–239, 1983.
23. [23] T.G. Lavers, Presentations of general products of monoids, J. Algebra 204, 733–741, 1998.
24. [24] S. MacLane, Homology, Classics in Mathematics, Springer Verlag, 1975.
25. [25] J.D.P. Meldrum, Wreath Products of Groups and Semigroups, Monographs and Sur- veys in Pure and Applied Mathematics (Book 74), Chapman and Hall/CRC; First Edition, 1995.
26. [26] P.W. Michor, Knit products of graded Lie algebras and groups, Rend. Circ. Mat. Palermo (2), 22, 171–175, 1989.
27. [27] J. Szép, On the structure of groups which can be represented as the product of two subgroups, Acta Sci. Math. Szeged, 12, 57–61, 1950.
28. [28] G. Zappa, Sulla costruzione dei gruppi prodotto di due sottogruppi permutabili tra loro, in: Atti Secondo Congresso Un. Ital., Bologna 1940. Edizioni Rome: Cremonense, 119–125, 1942.