# QUASI-COMMUTATIVE SEMIGROUPS OF FINITE ORDER RELATED TO HAMILTONIAN GROUPS

• Published : 2015.01.31
• 22 3

#### Abstract

If for every elements x and y of an associative algebraic structure (S, ${\cdot}$) there exists a positive integer r such that $ab=b^ra$, then S is called quasi-commutative. Evidently, every abelian group or commutative semigroup is quasi-commutative. Also every finite Hamiltonian group that may be considered as a semigroup, is quasi-commutative however, there are quasi-commutative semigroups which are non-group and non commutative. In this paper, we provide three finitely presented non-commutative semigroups which are quasi-commutative. These are the first given concrete examples of finite semigroups of this type.

#### Keywords

quasi-commutativity;finitely presented semigroups

#### References

1. C. M. Campbell, E. F. Robertson, N. Ruskuc, and R. M. Thomas, Semigroup and group presentations, Bull. Lond. Math. Soc. 27 (1995), no. 1, 46-50. https://doi.org/10.1112/blms/27.1.46
2. C. M. Campbell, E. F. Robertson, N. Ruskuc, R. M. Thomas, and Y. Unlu, Certain one-relator products of semigroups, Comm. Algebra 23 (1995), no. 14, 5207-5219. https://doi.org/10.1080/00927879508825527
3. M. Chacron and G. Thierrin, ${\sigma}$-reflexive semigroups and rings, Canad. Math. Bull. 15 (1972), 185-188. https://doi.org/10.4153/CMB-1972-033-3
4. A. H. Clifford and G. B. Preston, The Algebraic Theory of Semigroups I, Amer. Math. Soc., 1961.
5. A. C. Spoletini and A. Varisco, Quasicommutative semigroups and ${\sigma}$-reflexive semigroups, Semigroup Forum 19 (1980), no. 4, 313-321.
6. A. C. Spoletini and A. Varisco, Quasi Hamiltonian semigroups, Czechoslovak Math. J. 33 (1983), no. 1, 131-140.
7. J. M. Howie, An Introduction to Semigroup Theory, Academic Press Inc., 1976.
8. N. P. Mukherjee, Quasicommutative semigroups. I, Czechoslovak Math. J. 22 (1972), 449-453.
9. B. Pondelicek, Note on Quasi-Hamiltonian semigroups, Casopis pro pestovani matematiky 110 (1985), no. 4, 356-358.
10. K. P. Shum and X. M. Ren, On super Hamiltonian semigroups, Czechoslovak Math. J. 54 (2004), no. 1, 247-252. https://doi.org/10.1023/B:CMAJ.0000027264.87722.a8
11. K. P. Shum and L. Zhang, Generalized Quasi Hamiltonian semigroups, Int. J. Pure Appl. Math. 53 (2009), no. 4, 461-475.