Semigroup Construction on Polygonal Numbers

Abstract

In this paper, some information about polygonal numbers are given. Also, a general binary operator that includes all polygonal numbers are given and it is investigated whether the algebraic structures defined with the general operator specify a semigroup or not.

Some Algebraic Structure on Figurate Numbers

“…Specifically, it was proven by Sparavigna that it is a groupoid with binary operators defined on some polygonal numbers in [3]- [5]. Also, Emin studied semigroup construction on polygonal numbers in [6]. By using methods similar to those in these papers, we will give a general binary operator that includes all centered polygonal numbers.…”

“…And so, we obtain Starting from number 𝐶𝑆 5 (1) = 1, we have 6,16,31,51,76,91,106,141,181,226,276,331,391, … which are the elements of the set of 𝐵. From lemma 3.1.1. and theorem 3.1.1., one can say that the algebraic structure (𝐵, 𝛻) is a groupoid and semigroup.…”

“…As seen in Figure 3, centered polygonal numbers start from 𝐶𝑆 𝑚 (1) = 1. However, in some studies, as you can see in [1] and [6], centered polygonal numbers start from the number 𝐶𝑆 𝑚 (0) = 0. Now by considering the start point as the number 𝐶𝑆 𝑚 (0) = 0 and theorem 3.1.1., we can give the following corollary which gives the conditions for (𝐶, 𝛻) to be a monoid.…”