No Access

On paramedial division groupoids

Sergey Davidov

Department of Mathematics and Mechanics, Yerevan State University, Alek Manoogian 1, Yerevan 0025, Armenia

https://doi.org/10.1142/S179355711650008XCited by:3 (Source: Crossref)

Abstract

In this paper, we prove that regular paramedial division groupoids and binary regular paramedial division algebras are linear over an abelian group. Using this results we prove that every finitely generated paramedial division groupoid is a quasigroup and that every paramedial division groupoid is a homomorphic image of a paramedial quasigroup.

Communicated by N. Gilbert

Keywords:

AMSC: 20N05, 08A05

References

1. J. Cho, J. Ježek and T. Kepka, Paramedial groupoids, Czechoslovak Math. J. 49(2) (1999) 227–240. Web of Science, Google Scholar
2. S. S. Davidov, On regular medial division algebras, Quasigroups Related Systems 21 (2013) 155–164. Google Scholar
3. A. Ehsani and Yu. M. Movsisyan, Linear representation of medial-like algebras, Commun. Algebra 41(9) (2013) 3429–3444. Web of Science, Google Scholar
4. J. Ježek and T. Kepka, A note on medial division groupoids, Proc. Amer. Math. Soc. 119 (1993) 423–426. Web of Science, Google Scholar
5. J. Ježek and T. Kepka, Medial groupoids, in Monograph of Acad. Praha (1983). Google Scholar
6. J. Ježek and T. Kepka, The equational theory of paramedial cancellation groupoids, Czechoslovak Math. J. 50(1) (2000) 25–34. Web of Science, Google Scholar
7. Yu. M. Movsisyan, Hyperidentities in algebras and varieties, Russian Math. Surveys 53(1) (1998) 57–108. Web of Science, Google Scholar
8. P. Němec and T. Kepka, $T$ $T$ -quasigroups I, Acta Univ. Carolin. Math. Phys. 12(1) (1971) 39–49. Google Scholar
9. V. A. Shcherbakov and P. I. Pushkashu, On the structure of finite paramedial quasigroups, Comment. Math. Univ. Carolin. 51(2) (2010) 357–370. Google Scholar
10. A. Kh. Tabarov, Homomorphisms and endomorphisms of linear and alinear quasigroups, Discrete Math. Appl. 17(3) (2007) 235–260. Google Scholar