ON CONNECTEDNESS AND COMPLETENESS OF CAYLEY DIGRAPHS OF TRANSFORMATION SEMIGROUPS WITH FIXED SETS

Year 2020, Volume: 28 Issue: 28, 110 - 126, 14.07.2020

Nuttawoot Nupo Chollawat Pookpıenlert

https://doi.org/10.24330/ieja.768190

Cited By: 3

Abstract

Let $\text{Fix}(X,Y)$ be a semigroup of full transformations on a set $X$ in which elements in a nonempty subset $Y$ of $X$ are fixed. In this paper, we construct the Cayley digraphs of $\text{Fix}(X,Y)$ and study some structural properties of such digraphs such as the connectedness and the completeness. Further, some prominent results of Cayley digraphs of $\text{Fix}(X,Y)$ relative to minimal idempotents are verified. In addition, the characterization of an equivalence digraph of the Cayley digraph of $\text{Fix}(X,Y)$ is also investigated.

Keywords

Cayley digraphs of transformation semigroups, connectedness, completeness, minimal idempotents, equivalence digraphs

References

• M. Afkhami, H. Reza Barani, K. Khashyarmanesh and F. Rahbarnia, A new class of Cayley graphs, J. Algebra Appl., 15(4) (2016), 1650076 (8 pp).
• Y. Chaiya, P. Honyam and J. Sanwong, Natural partial orders on transformation semigroups with fixed sets, Int. J. Math. Math. Sci., (2016), 2759090 (7 pp).
• A. H. Clifford and G. B. Preston, The Algebraic Theory of Semigroups, Vol. I, Mathematical Surveys, 7, American Mathematical Society, Providence, R.I., 1961.
• A. H. Clifford and G. B. Preston, The Algebraic Theory of Semigroups, Vol. II, Mathematical Surveys, 7, American Mathematical Society, Providence, R.I., 1967.
• S. Fan and Y. Zeng, On Cayley graphs of bands, Semigroup Forum, 74(1) (2007), 99-105.
• Y. Hao and Y. Luo, On the Cayley graphs of left (right) groups, Southeast Asian Bull. Math., 34(4) (2010), 685-691.
• P. Honyam and J. Sanwong, Semigroups of transformations with fixed sets, Quaest. Math., 36(1) (2013), 79-92.
• J. M. Howie, Fundamentals of Semigroup Theory, London Mathematical Society Monographs, New Series, 12, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.
• A. V. Kelarev, On Cayley graphs of inverse semigroups, Semigroup Forum, 72(3) (2006), 411-418.
• B. Khosravi and M. Mahmoudi, On Cayley graphs of rectangular groups, Discrete Math., 310(4) (2010), 804-811.
• Y. Luo, Y. Hao and G. T. Clarke, On the Cayley graphs of completely simple semigroups, Semigroup Forum, 82(2) (2011), 288-295.
• R. P. Panda and K. V. Krishna, On connectedness of power graphs of finite groups, J. Algebra Appl., 17(10) (2018), 1850184 (20 pp).
• S. Pirzada, An Introduction to Graph Theory, Universities Press, Orient Blackswan, India, 2012.
• A. Riyas and K. Geetha, Some properties of Cayley graphs of full transformation semigroups, Int. J. Math. Archive, 9(4) (2018), 64-69.
• D. Sinha and D. Sharma, Structural properties of absorption Cayley graphs, Appl. Math. Inf. Sci., 10(6) (2016), 2237-2245.
• T. Suksumran and S. Panma, On connected Cayley graphs of semigroups, Thai J. Math., 13(3) (2015), 641-652.
• C. Tisklang and S. Panma, On connectedness of Cayley graphs of finite transformation semigroups, Thai J. Math., Special issue (2018), 261-271.