KaydedilenlerDerginizi ekleyin

Dergiler / Hacettepe Journal of Mathematics and Statistics / 2021 / Cilt: 50 - Sayı: 1

A higher version of Zappa products for monoids

Ahmet Sinan Çevik
Suha Ahmad Wazzan
Fırat Ateş
Sayfa
224–234
Yayın Tarihi
2022-10-18
DOI
—
🏛️Bu makalenin bibliyografik verileri ve tam metin bağlantısı DergiPark üzerinden sağlanmıştır.
DergiPark Kaydı ↗
✨

Yapay Zeka Hızlı Bakış & Çıkarımlar

AI Insight
⚡ TL;DR(1 Cümlede Hızlı Çıkarım)

At the final part of this paper we point out some possible future problems related to this subject.

📌 3 Maddede Bu Makale

🎯
Amaç & Hipotez: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.
📊
Temel Bulgular & Veri: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 $A$.
💡
Sonuç & Katkı:At the final part of this paper we point out some possible future problems related to this subject.
🔬Tür:Araştırma Makalesi
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:Knit products · Wreath products · Zappa products

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.

🔖

Bu Makaleye Atıf Yap

Çevik, A. S., Wazzan, S. A., Ateş, F. (2022). A higher version of Zappa products for monoids. Hacettepe Journal of Mathematics and Statistics, Cilt: 50 - Sayı: 1, 224–234. https://ajindex.com/articles/529952

📄 Tam Metin PDF Okuyucu

📄

PDF görüntüleyici hazırlanıyor...

Kaynakça (18)

  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. 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. 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. H. Ayik, F. Kuyucu and B. Vatansever, On semigroup presentations and efficiency, Semigroup Forum, 65, 329–335, 2002
  5. A. Ballester-Bolinches, E. Cosme-Llopez and R. Esteban-Romero, Group extensions and graphs, Expo. Math. 34 (3), 327–334, 2016
  6. A. Ballester-Bolinches, R. Esteban-Romero and M. Asaad, Products of Finite Groups, de Gruyter Exp. Math. 53, Walter de Gruyter, 2010
  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. 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. 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. G. Baumslag, Wreath products and finitely presented groups, Math. Z. 75, 22–28, 1961
  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. 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. M.G. Brin, On the Zappa-Szép product, Comm. Algebra 33, 393–424, 2005
  14. A.S. Cevik, The efficiency of standard wreath product, Proc. Edinburgh Math. Soc. 43 (2), 415–423, 2000
  15. A.S. Cevik, Minimal but inefficient presentations of the semi-direct product of some monoids, Semigroup Forum, 66 (1), 1–17, 2003
  16. N.D. Gilbert and S. Wazzan, Zappa-Szép products of bands and groups, Semigroup Forum, 77, 438–455, 2008
  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. J.M. Howie and N. Rus