Some Groupoids and their Representations by Means of Integer Sequences

 + 39-011-090-7360 This article is published under the terms of the Creative Commons Attribution License 4.0 Author(s) retain the copyright of this article. Publication rights with Alkhaer Publications. Published at: http://www.ijsciences.com/pub/issue/2019-10/ DOI: 10.18483/ijSci.2188; Online ISSN: 2305-3925; Print ISSN: 2410-4477 Some Groupoids and their Representations by Means of Integer Sequences Amelia Carolina Sparavigna


Introduction
A groupoid is an algebraic structure made by a set with a binary operator [1]. The only restriction on the operator is closure. This property means that, applying the binary operator to two elements of a given set S, we obtain a value which is itself a member of S. If this binary operation is associative and we have a neutral element and opposite elements into the set, the groupoid becomes a group.
Groupoids are interesting also for the study of integer numbers. As shown in some previous works [2 -7], the integer sequences of Mersenne, Fermat, Cullen, Woodall and other numbers are groupoid possessing different binary operators. Here we show that other integer sequences can have the same binary operators, and therefore can be used to represent the related groupoids. That is, we can obtain different integer sequences by means of the recurrence relations generated by the considered binary operations.
In [7], we started the search for different representations for the groupoid of Triangular Numbers. Here we generalize this search, using the binary operators obtained in the previous analyses. In particular, we will see the representations linked to Mersenne, Fermat, Cullen, Woodall, Carol and Kynea, and Oblong numbers. The binary operators of these numbers have been already discussed in previous works. The results concerning the Triangular numbers are also reported.
Using the On-Line Encyclopedia of Integer Sequences (OEIS), we are able to identify the several representations of groupoids. At the same time, we can also find integer sequences not given in OEIS and probably not yet studied.

Mersenne numbers
We discussed the binary operator of the set of Mersenne numbers in [8,9]. The numbers are given as . The binary operator is: As shown in [9], this binary operation is a specific case of the binary operator of q-integers, which can be linked to the generalized sum of Tsallis entropy [10,11]. The binary operator can be used to have a recurrence relation: (2) Here in the following, let us show the sequences that we can generate from (1) and (2). We use OEIS, the On-Line Encyclopedia of Integer Sequences, to give more details on them.

Fermat numbers
The group of Fermat numbers has been discussed in [13]. As explained in [14], there are two definitions of the Fermat numbers. "The less common is a number of the form obtained by setting x=1 in a Fermat polynomial, the first few of which are 3, 5, 9, 17, 33, ... (OEIS A000051)" [14]. We used this definition.
The binary operator can be used to have a recurrence relation: Sequences can generate from (3) and (4).

Cullen and Woodall numbers
These numbers had been studied in [15]. Let us consider the Cullen numbers, . We have the binary operator: , Woodall numbers are , and the binary operator is:

Carol and Kynea Numbers
These numbers have been studied in [3]. Carol number is: ( ) . The binary operator is given in [3]: Let us consider the Kynea numbers.

( )
The binary operator is given in [3]. We use again . Again, we have square roots, so we can obtain integer sequences only in some cases.
Of course, we can repeat the same approach for the odd squares (A016754) numbers. Their binary operator is given in [4]. Also for the centered square numbers and the star numbers, we have the binary operators [5,6], so we can find the related representations by means of integer sequences too. As previously told, among the generated sequences, news sequences are produced that can be interesting for further investigation of integer sequences.

Triangular numbers
These numbers are really interesting. The numbers are of the form (OEIS A000217):

∑ ( )
I have discussed them in [7]. For these numbers we can give two binary operators. For the convenience of the reader, I show the results that we can obtain. The first binary operator is [7]: Again we consider , and change the value of . Here in the following the sequences that we generate.
As previously told, we have a second binary operator for the triangular numbers [7]. It is the following: Again, let us consider as we did before. Using the On-Line Encyclopedia of Integer Sequences (OEIS), we have seen that quite different sequences can have the same binary operators. We have also found integer sequences not given in OEIS and that need to be studied.

Conclusion
Groupoids are related to the integer sequences. These groupoid possess different binary operators. As we have shown, other integer sequences can have the same binary operators, and therefore can be used to represent the related groupoids.