Cayley hypergraph over polygroups

Document Type : Research Paper


Department of Mathematical Sciences, Yazd University, Yazd, Iran



Comer introduced a class of hypergroups, using the name of polygroups. He emphasized the importance of polygroups, by analyzing them in connections to graphs, relations, Boolean and cylindric algebras. Indeed, polygroups are multi valued systems that satisfy group like axioms. Given a polygroup with a finite generating set, we can form a Cayley hypergraph for that polygroup with respect to that generating set. This helps us to better understand and investigate polygroup structures. More precisely,
in this paper, we introduce the construction of Cayley hypergraphs over polygroups, say $CH(\mathbf{P},S)$ such that $\mathbf{P}$ is a polygroup and $\langle S\rangle =P$. We investigate some properties of them. It is well known to give a constructing for building a big polygroup from two small ones. This structure is called extension
of polygroups. In particular, we describe the connection between Cayley hypergraphs over extension of two polygroups and Cartesian product of two Cayley hypergraphs.


Main Subjects

[1] C. Berge, Graphs and Hypergraphs, 6, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973.
[2] A. Bretto, Hypergraph Theory. An introduction, Mathematical Engineering, Springer, Cham, 2013.
[3] M. Buratti, Cayley, Marty and Schreier hypergraphs, Abh. Math. Sem. Univ. Hamburg, 64 (1994) 151–161.
[4] S. D. Comer, Polygroups derived from cogroups, J. Algebra, 89 no. 2 (1984) 397–405.
[5] P. Corsini, Hypergraphs and hypergroups, Algebra Universalis, 35 no. 4 (1996) 548–555.
[6] B. Davvaz, Polygroup theory and related systems, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013.
[7] M. Farshi, B. Davvaz and S. Mirvakili, Degree hyoergroupoids associated with hypergraphs, Filomat, 28 no. 1 (2014) 119–129.
[8] A. Iranmanesh and M. N. Iradmusa, The combinatorial and algebraic structure of the hypergroup associated to a hypergraph, . Mult.-Valued Logic Soft Comput., 11 no. 1-2 (2005) 127–136.
[9] J. Lee and Y. S. Kwon, Cayley hypergraphs and Cayley hypermaps, Discrete Math., 313 no. 4 (2013) 540–549.
[10] V. Leoreanu and L. Leoreanu, Hypergroups associated with hypergraphs, Ital. J. Pure Appl. Math., 4 (1998) 119–126.
[11] F. Marty, Sur une generalization de la notion de groupe, In: 8th congres Math. Scandinaves, Stockholm, (1934) 45–49.
[12] J. Meier, Groups Graphs and Trees: An introduction to the Geometry of infinite groups, Cambridge University Press, Cambridge, 2008.
[13] A. Nikkhah, B. Davvaz and S. Mirvakili, Hypergroups constructed from hypergraphs, Filomat, 32 no. 10 (2018) 3487–3494.

Articles in Press, Corrected Proof
Available Online from 20 January 2024
  • Receive Date: 02 November 2023
  • Revise Date: 28 December 2023
  • Accept Date: 20 January 2024
  • Published Online: 20 January 2024