Some Algebraic Structure on Figurate Numbers

In this study, some information about figurate numbers and centered polygonal numbers are given. Also, a general binary operator that includes all centered polygonal numbers is defined and it is investigated whether the algebraic structures defined with the general binary operator specify a groupoid and semigroup or not. And finally, some examples are given on the subject.

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[1] E. Deza and M. M. Deza, Figurate Numbers; World Scientific Publishing Company, Singapore, 2014.
[2] T.L. Heath, Diophantus of Alexandria: A study in the History of Greek Algebra; CUP Archive, 1910.
[3] A. C. Sparavigna, “Groupoids of OEIS A003154 Numbers (star numbers or centered dodecagonal numbers),” Zenodo, doi: 10.5281/zenodo.3387054, 2019.
[4] A. C. Sparavigna, “Groupoids of OEIS A093112 and A093069 Numbers (oblong and odd square numbers),” Zenodo, doi: 10.5281/zenodo.3247003, 2019.
[5] A. C. Sparavigna, “On a generalized sum of the Mersenne Numbers,” Zenodo, doi: 10.5281/zenodo.2634312, 2018.
[6] A. Emin, “Semigroup Construction On Polygonal Numbers,” Journal of Engineering Technology and Applied Sciences., 2021.
[7] A. Emin, F. Ateş, “Some New Results on the Orthodox, Strongly π- Inverse and π- regularity of Some Monoids,” Bulletin of The Society of Mathematicians Banja Luka, vol. 11, no. 3, 2021.
[8] A. Emin, F. Ateş, S. Ikikardeş, and I. N. Cangül, “A new monoid construction under crossed products,” Journal of Inequalities and Applications., vol. 2013, no. 1, 2013.
[9] F. Ateş, “Some new monoid and group constructions under semidirect products,” Ars. Combinatoria, vol. 2009, no. 1, 2009.
[10] E. K. Cetinalp, “Regularity of Iterated Crossed Product of Some Monoids,” Bulletin of The Society of Mathematicians Banja Luka, vol. 12, no. 1, 2022.
[11] J. M. Howie, Fundamentals of semigroup theory. Oxford, England: Clarendon Press, 1995.
[12] E. W. Weisstein, “Groupoid,” https://mathworld.wolfram.com/. [Accessed: 24-Mar-2022].
[13] “The on-line encyclopedia of integer sequences® (OEIS®),” Oeis.org. [Online]. Available: http://oeis.org. [Accessed: 24-Mar-2022].