Diagonal function of natural rhotrix

Abstract

Natural rhotrix refers to the rhotrix whose elements are all natural numbers, arrayed in their natural order. This type of rhotrix has just recently been introduced in literature. Therefore, this work is taking a further look at its properties. It introduces the concept of diagonal function of a natural rhotrix and examines its properties. It is found that the diagonal function $d(A_n)$ of a natural rhotrix $R_n$ is given as $d(A_n)=(2\rho +1)h(R_n)$. Furthermore, it presents the sum of the other elements outside its diagonals as $d(A_n^c)=(|R_n|+1-2n)h(R_n)$ where $A_n$ is the set of elements along any of its diagonals, $A_n^c$, $\rho$ and $h(R_n)$ are the complement of $A_n$, the index and the heart of $R_n$ respectively. These properties are all peculiar to this beautiful set of rhotrix called the natural rhotrix.

Keywords:

• natural rhotrix
• n-dimensional
• diagonal function
• heart and index

PUBLIC INTEREST STATEMENT

In engineering, architecture, building and arts, there is a paradigm shift from rectangular to rhomboidal designs and shapes. Most halls, conference centers and lecture theaters are designed to be rhomboidal in shapes and structures. Such structures have two long diagonals that are equal in length, and the interception of these diagonals is the heart. The concern of structural engineers and architects is the capacity of the event centers or conference halls they design.

This work is a scientific study of the rhomboidal structures. For example, the number of seats occupying any of the diagonals is the same, and this determines the capacity of the Hall. And the position of the middle seat at the diagonal plays a major role in determining the capacity of the Hall. Thus, using appropriate mathematical links in the work one can determine at a glance the capacity of the designed structures.

1. Introduction

The representation of numbers in a geometric shape could be traced to the work of James Joseph Sylvester in $1848$, who represented numbers in a rectangular shape. This representation was called a matrix. Since then, representation of numbers in different forms has been of interest to researchers. For example, Atanassov and Shannon (1998) gave a mathematical array that lies, in some way, between two-dimensional vectors and $(2\times 2)$-dimensional matrices, and denoted it as matrix-tertions and noitrets. Extending this idea, Ajibade (2003), introduced objects which are, in some ways, between $(2\times 2)$ and $(3\times 3)$-dimensional matrices. He called such an object a rhotrix. Thus, he defined a rhotrix $R$ of dimension three as a rhomboidal array given as:

$R=⟨\begin{array}{ccc}\hspace{0.17em}& a& \hspace{0.17em}\\ b& c& d\\ \hspace{0.17em}& e& \hspace{0.17em}\end{array}⟩$

where $a,b,c,d,e\in \mathrm{\Re }$. The entry $c$ is called the heart of $R$, denoted as $h(R)$. Since its birth in $2003$, many researchers have shown interest in developing and expanding this concept, most often, in analogy with the concepts of matrices usually through a transformation that converts a matrix into a rhotrix and vice versa as earlier remarked in the maiden manuscript-see Ajibade (2003). One of such works was the classification of rhotrices into sets and algebraic spaces by Mohammed and Tella (2012). The paper classifies rhotrices into natural rhotrix set, real rhotrix set, complex rhotrix set, rational and irrational rhotrix sets, and thus, opening up different branches of studying rhotrices. In $2004$, Sani (2004) proposed the first alternative rhotrix multiplication called the row-column multiplication, and was generalized in $2007$—Sani (2004, 2007). He also reported that when an odd-dimensional matrix is rotated clockwise about ${45}^o$, a rhotrix is obtained, and rotating a rhotrix anti-clockwise through the same angle, a special matrix called a coupled matrix is obtained (see Sani (2008)). Therefore, every rhotrix is a coupled matrix, and vice versa. Most times, when we refer to rhotrices, we are seeing them as coupled matrices. That explains why we rightly refer to the major column and row of a rhotrix as its major and minor diagonals, respectively. Using this special matrix of dimension $n$, a two-different systems of linear equations was solved simultaneously where one is an $(n\times n)$ system $AX=b$ while the other is an $(n-1)\times (n-1)$ system, $CY=d$ - (see Sani (2009)). A system of linear equations arising from the rhotrix equation $A\odot X=C$ was examined in Aminu (2009).

Definition 1.1. (Mohammed & Tella, 2012). A rhotrix set is called a natural rhotrix set if its entries belong to the set of natural numbers. For example,

${\hat{R}}_3(\mathbb{N})=\left\{⟨\begin{array}{ccc}\hspace{0.17em}& a& \hspace{0.17em}\\ b& c& d\\ \hspace{0.17em}& e& \hspace{0.17em}\end{array}⟩:a,b,c,d,e\in \mathbb{N}\right\}$

is the set of all three dimensional natural rhotrices.

Remark 1.1 Recall that the set $\mathbb{N}=\{0,1,2,3,\dots \}$ is the set of natural numbers or better still the set of non-negative integers- see Aashikpelokhai et al. (2010) and Baumslag and Chandler (1968). Thus, the set $2\mathbb{N}+1$ is larger than $2{\mathbb{Z}}^{+}+1$.

A formal definition of a real rhotrix as presented in the original paper is given below:

Definition 1.2. Ajibade (2003), and Mohammed and Tella (2012). A real rhotrix set of dimension three, denoted as ${\hat{R}}_3(\mathrm{\Re })$ is defined as:

${\hat{R}}_3(\mathrm{\Re })=\left\{⟨\begin{array}{ccc}\hspace{0.17em}& a& \hspace{0.17em}\\ b& c& d\\ \hspace{0.17em}& e& \hspace{0.17em}\end{array}⟩:a,b,c,d,e\in \mathrm{\Re }\right\}$

where the entry $c$ is called the heart of any rhotrix $R[h(R)]$ belonging to ${\hat{R}}_3(\mathrm{\Re })$ and $\mathrm{\Re }$ is the set of real numbers.

Rhotrix theory is still relatively new and many researches are on-going in the field. Mohammed (2009) classified rhotrix as abstract structures preparing the stage for serious works in rhotrix algebra. Shortly, Tundunkaya and Manjuola (2010) constructed rhotrix finite fields. Afterward, Ezugwu et al. (2011) worked together to generalize heart-based multiplication of rhotrix to n-dimensional rhotrix. A remarkable effort was made by Mohammed et al. (2012) to represent a rhotrix over a linear map. Usaini and Mohmmed (2012) investigated rhotrix eigenvalues and eigenvectors for possible solutions to a system of linear equations. Furthermore, Mohammed (2014) introduced a new expression for rhotrix representation. The first review of articles on rhotrix theory during the first decade was presented in Mohammed and Balarabe (2014). It is worthy to note that an n-dimensional heart-based rhotrix denoted by ${\hat{R}}_n(\mathrm{\Re })$ has its cardinality as $|{\hat{R}}_n(\mathrm{\Re })|=\frac{1}{2}(n^2+1)$, where $n\in 2{\mathbb{Z}}^{+}+1$. This implies that all heart-based or heart-oriented rhotrices are of odd dimension ($\ge 3$). Aminu and Michael (2015) made a strenuous effort to represent an array of numbers in form of a parallelogram. They called this representation a paraletrix. This representation generalizes rhotrix since it allows the length of the structures to vary. Besides, it was observed that not all paraletrix have a heart. This was a motivation for Isere (2017, 2018, 2019) to investigate a rhotrix without a heart. Representing a rhotrix this way helps to define an even-dimensional rhotrix. All these works were chronicled as associative rhotrix theory in Isere and Adeniran (2018), and presented the non-associative binary multiplication of rhotrix sets. The addition and multiplication of two heart-based rhotrices were defined in Ajibade (2003) as:

$R+Q=⟨\begin{array}{ccc}\hspace{0.17em}& a& \hspace{0.17em}\\ b& h(R)& d\\ \hspace{0.17em}& e& \hspace{0.17em}\end{array}⟩+⟨\begin{array}{ccc}\hspace{0.17em}& f& \hspace{0.17em}\\ g& h(Q)& j\\ \hspace{0.17em}& k& \hspace{0.17em}\end{array}⟩=⟨\begin{array}{ccc}\hspace{0.17em}& a+f& \hspace{0.17em}\\ b+g& h(R)+h(Q)& d+j\\ \hspace{0.17em}& e+k& \hspace{0.17em}\end{array}⟩$

and

$R\circ Q=⟨\begin{array}{ccc}\hspace{0.17em}& ah(Q)+fh(R)& \hspace{0.17em}\\ bh(Q)+gh(R)& h(R)h(Q)& dh(Q)+jh(R)\\ \hspace{0.17em}& eh(Q)+kh(R)& \hspace{0.17em}\end{array}⟩$

respectively. A generalization of this hearty multiplication is given in Mohammed (2014) and in Ezugwu et al. (2011). A row-column multiplication of heart-based rhotrices was proposed by Sani (2004) as:

$R\circ Q=⟨\begin{array}{ccc}\hspace{0.17em}& af+dg& \hspace{0.17em}\\ bf+eg& h(R)h(Q)& aj+dk\\ \hspace{0.17em}& bj+ek& \hspace{0.17em}\end{array}⟩$

A generalization of this row-column multiplication was also later given by Sani (2007) as:

$R_n\circ Q_n=⟨a_{i_1{j}_1},c_{i_2{j}_2}⟩\circ ⟨b_{i_2{j}_2},d_{l_2{k}_2}⟩=⟨\sum _{i_2{j}_1=1}^t(a_{i_1{j}_1}{b}_{i_2{j}_2}),\sum _{l_2{k}_1=1}^{t-1}(c_{l_1{k}_1}{d}_{l_2{k}_2})⟩,t=(n+1)/2.$

where $R_n$ and $Q_n$ are $n$-dimensional rhotrices (with $n$ rows and $n$ columns). Commenting on this method of multiplication by Mohammed and Tella (2012) says: “a unique expression for the ‘rhotrix-heart’ cannot be deduced and therefore, this method of rhotrix expression is unsuitable for presenting heart-based rhotrices”. This challenge was addressed by the introduction of even-dimensional rhotrices where this multiplication method can fit in naturally—See Isere (2017, 2018, 2019).

2. Preliminaries

In this section, some definitions that are peculiar to the natural rhotrix will be considered. Some fundamental results as discussed in Isere (2016) will be presented. These results are helpful in the computations to follow.

Definition 2.1. Isere (2016)(Major row and major column)

The major row and the major column are usually the only full row and full column in a rhotrix. They are normally at the middle of the rows and columns of any dimensional rhotrix. These are also called the major and minor diagonals of the rhotrix.

Definition 2.2. An $n$-dimensional natural rhotrix $(R_n)$ is generalized as:

$R_n=⟨\begin{array}{ccccccc}\hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}& 1& \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}\\ \hspace{0.17em}& \hspace{0.17em}& 2& 3& 4& \hspace{0.17em}& \hspace{0.17em}\\ \hspace{0.17em}& -& -& -& -& -& \hspace{0.17em}\\ m^2+1& -& -& m^2+m+1& -& -& m^2+2m+1\\ \hspace{0.17em}& -& -& -& -& -& \hspace{0.17em}\\ \hspace{0.17em}& \hspace{0.17em}& 2m^2+2m-2& 2m^2+2m-1& 2m^2+2m& \hspace{0.17em}& \hspace{0.17em}\\ \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}& 2m^2+2m+1& \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}\end{array}⟩$

where $1,2,3,\dots ,2m^2+2m+1\in \mathbb{N}\mathrm{\forall }n\in 2\mathbb{N}+1$ and $m\in \mathbb{N}$ (i.e. $n=2m+1$)

Remark 2.1. The above is the generalization of any natural rhotrix with ascending ordered entries. Note also that all our entries are non-zero elements of $\mathbb{N}$. With this generalization, one can represent a natural rhotrix of any dimension at a glance.

2.1. Properties of natural rhotrix

This subsection presents some properties of this unique rhotrix. These properties are fundamental to the main section. They are presented in Isere (2016).

Lemma 2.1. Let $R_i$ be any $i$ -dimensional natural rhotrix. Then, the heart $(h(R))$ is the middle value of a set of $n$ numbers that make up the rhotrix if and only if $n=|R_i|i=1,3,5,\dots$

Theorem 2.1. Let $R_n$ be any $n$-dimensional natural rhotrix. Then, the following are equivalent:

(a) The cardinality $|R_n|=\frac{1}{2}(n^2+1)$ where $n\in 2\mathbb{N}+1$

(b) The last entry gives the value $2m^2+2m+1\mathrm{\forall }m\in \mathbb{N}$

(c) The heart of $R_n,h(R_n)$ is represented by $h=\frac{1}{2}(|R_n|+1),n\in 2\mathbb{N}+1$

(d) The $h(R_n)$ is the value $m^2+m+1$

Proposition 2.1. The determinant function of a heart-based rhotrix is its heart

Remark 2.2. Though natural rhotrices are singular rhotrices, and are not invertible, the heart, however performs a determinant function depending on the dimension of the rhotrix.

Evaluating the determinant of a natural rhotrix $R$ is simply the value of its heart ($h(R)$). However, a rhotrix of higher dimension is characteristically not only being determined by its heart. There are other entries in the neighbourhood of the heart that also help in determining the true nature of a natural rhotrix. That brings us to the next subsection.

2.2. Codeterminants

The concept of the codeterminant was also introduced in Isere (2016). This concept is peculiar to rhotrices with ordered entries. We shall adopt $codet(A)$ to mean the codeterminant of a rhotrix $A$.

Let consider an example

Example 2.1. Find the codeterminant of the natural rhotrix below (a) along major column (b) along the major row of the rhotrix (see Isere (2016)).

$Q=⟨\begin{array}{ccccccc}\hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}& 1& \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}\\ \hspace{0.17em}& \hspace{0.17em}& 2& 3& 4& \hspace{0.17em}& \hspace{0.17em}\\ \hspace{0.17em}& 5& 6& 7& 8& 9& \hspace{0.17em}\\ 10& 11& 12& 13& 14& 15& 16\\ \hspace{0.17em}& 17& 18& 19& 20& 21& \hspace{0.17em}\\ \hspace{0.17em}& \hspace{0.17em}& 22& 23& 24& \hspace{0.17em}& \hspace{0.17em}\\ \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}& 25& \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}\end{array}⟩$

Solution

(a) $codet(Q)$ along the major column

$codet(Q)=⟨\begin{array}{ccc}\hspace{0.17em}& 1& \hspace{0.17em}\\ 2& 3& 4\\ \hspace{0.17em}& 7& \hspace{0.17em}\end{array}⟩+⟨\begin{array}{ccc}\hspace{0.17em}& 7& \hspace{0.17em}\\ 12& 13& 14\\ \hspace{0.17em}& 19& \hspace{0.17em}\end{array}⟩+⟨\begin{array}{ccc}\hspace{0.17em}& 19& \hspace{0.17em}\\ 22& 23& 24\\ \hspace{0.17em}& 25& \hspace{0.17em}\end{array}⟩=3+13+23=39$

(b) $codet(Q)$ along major row

$codet(Q)=⟨\begin{array}{ccc}\hspace{0.17em}& 5& \hspace{0.17em}\\ 10& 11& 12\\ \hspace{0.17em}& 17& \hspace{0.17em}\end{array}⟩+⟨\begin{array}{ccc}\hspace{0.17em}& 7& \hspace{0.17em}\\ 12& 13& 14\\ \hspace{0.17em}& 19& \hspace{0.17em}\end{array}⟩+⟨\begin{array}{ccc}\hspace{0.17em}& 9& \hspace{0.17em}\\ 4& 15& 16\\ \hspace{0.17em}& 21& \hspace{0.17em}\end{array}⟩=11+13+15=39$

Thus, the codeterminant values of $Q$ are $\{3,13,23\}$ (vertically) or $\{11,13,15\}$ (horizontally) depending on how you view the rhotrix $Q$.

Remark 2.3. Either of the two results (a) or (b) above is sufficient. Therefore, finding the determinant of a higher dimensional natural rhotrix is simply finding the sum of its codeterminant values. For example, the determinant of $Q$ above is $39$ which is $3\times h(Q)$.

Observe that the determinant function of the rhotrix $Q$ is simply $3\times h(Q)$, where the value $3$–the multiplier is the index of $Q$. That takes us to the next subsection.

2.3. Index of a natural rhotrix

Definition 2.3. The index of a rhotrix $A$ is the number of minor rhotrices of dimension three that can be derived, either along the major column or along the major row, from $A$. This index is a whole number or better still a natural number.

Remark 2.4. The index of a rhotrix $Q$ in Example 2.1 is $3$—the number of rhotrices of dimension three derived from $Q$. Therefore, the index of $R_3$ is $1$ and the indices of $R_5$, $R_7$ and $R_9$ are $2$, $3$ and $4$ respectively. Appropriately, the index of $R_1$ is zero.

Theorem 2.2. Given any rhotrix $R$, the $codet(R)=\rho det(R)$ where $\rho$ is a natural number called the index of $R$

In other words, the determinant function of any dimensional natural rhotrix is given as $det(R_n)=\rho h(R_n)$ where $n$ and $\rho$ are the dimension and the index of $R_n$ respectively.

Theorem 2.3. Giving any natural rhotrix $R$

$codet(R)=\frac{\rho }{2}(|R|+1)$

where $\rho$ is the index and $|R|$ is the cardinality of $R$, and $|R_n|=\frac{1}{2}(n^2+1)$

3. Main results

In this section, further properties of the natural rhotrix will be examined and presented.

Definition 3.1. Let $R_n$ be an $n$-dimensional natural rhotrix and $A_n=\{a_1,a_2,\dots ,a_n\}$ a set of ordered natural numbers along the major diagonal of $R_n$. If $d(A_n)$ denotes the sum of $a_1,a_2,\dots ,a_n$, then $d$ is a positive function on $R_n$ called the diagonal function.

Remark 3.1. The diagonal function $d(A_n)$ is a positive function and it corresponds to the trace of natural rhotrix [$tr(R_n)$]. Unlike the trace of a real rhotrix, the sum of elements of each of the diagonals of a natural rhotrix are equal. This will be discussed in the following result.

Lemma 3.1. Let $R_n$ be a natural rhotrix, and let $A_n=\{a_1,a_2,\dots ,a_n\}$ and $B_n=\{b_1,b_2,\dots ,b_n\}$ be the sets of ordered natural numbers along the major and minor diagonals of $R_n$ respectively. If $d(A_n)=a_1+a_2+\dots +a_n$ and $d(B_n)=b_1+b_2+\dots +b_n$ then $d(A_n)=d(B_n)$.

Proof:

Since the $|A_n|$ and $|B_n|$ are the cardinality of the major and minor diagonals of $R_n$, then $|A_n|=n=|B_n|$. Observe that the heart is the intersection of the two diagonals. Therefore, by Lemma 2.1 $h(R_n)=\frac{d(A_n)}{|A_n|}=\frac{d(B_n)}{|B_n|}$. Thus $d(A_n)=d(B_n)$ (since $|A_n|=|B_n|$).

Remark 3.2. Consider the examples: Find the $d(A_n)$ and $d(B_n)$ in the rhotrices below, where $n=3$ and $7$ respectively.

(i)

$R_3=⟨\begin{array}{ccc}\hspace{0.17em}& 1& \hspace{0.17em}\\ 2& 3& 4\\ \hspace{0.17em}& 5& \hspace{0.17em}\end{array}⟩$

(ii)

$R_7=⟨\begin{array}{ccccccc}\hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}& 1& \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}\\ \hspace{0.17em}& \hspace{0.17em}& 2& 3& 4& \hspace{0.17em}& \hspace{0.17em}\\ \hspace{0.17em}& 5& 6& 7& 8& 9& \hspace{0.17em}\\ 10& 11& 12& 13& 14& 15& 16\\ \hspace{0.17em}& 17& 18& 19& 20& 21& \hspace{0.17em}\\ \hspace{0.17em}& \hspace{0.17em}& 22& 23& 24& \hspace{0.17em}& \hspace{0.17em}\\ \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}& 25& \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}\end{array}⟩$

Solution

(i)

$d(A_3)=1+3+5=9$

and

$d(B_3)=2+3+4=9$

(ii)

$d(A_7)=1+3+7+13+19+23+25=91$

and

$d(B_7)=10+11+12+13+14+15+16=91$

Therefore, $d(A_n)=d(B_n)n=3,7$. This holds for all $n\in 2\mathbb{N}+1$

Theorem 3.1. Let $\mathbb{N}$ be a set of a natural numbers and ${\hat{R}}_n(\mathbb{N})$ be a set of natural rhotrices of dimension $n$. Then, there exists a bijective map $\rho :{\hat{R}}_n(\mathbb{N})\mapsto \mathbb{N}$ given as $\rho (R_n)=m\mathrm{\forall }m\in \mathbb{N}$ and $n=2m+1$

Proof:

We need to show that $\rho$ is both $1-1$ and onto map. Let $R_n$ and $R_n^{\ast }$ be in ${\hat{R}}_n(\mathbb{N})$. Suppose, $\rho (R_n)=m$ and $\rho (R_n^{\ast })=m^{\ast }$ for all $m,m^{\ast }\in \mathbb{N}$. Then $\rho (R_n)=\rho (R_n^{\ast })\iff m=m^{\ast }$ which implies that $R_n=R_n^{\ast }$. Thus $\rho$ is $1-1$. Since for every $m\in \mathbb{N}$ there exists $R_n\in {\hat{R}}_n(\mathbb{N})$, then $\rho (R_n)=\mathbb{N}$. Then $\rho$ is onto. Thus $\rho$ is bijective.

Remark 3.3. $\rho$ appropriately corresponds to the index of a natural rhotrix as presented in Isere (2016). For example, $\rho (R_1)=0$(trivial), $\rho (R_3)=1$, $\rho (R_5)=2$ etc.

Theorem 3.2. Let $R_n$ be any $n$-dimensional natural rhotrix, and if $\rho$ and $h(R_n)$ are the index and the heart of $R_n$ respectively, then the sum of elements along the major or minor diagonal of $R_n$ denoted as $d(A_n)$ is given as

$d(A_n)=(2\rho +1)h(R_n)\mathrm{\forall }n\in 2\mathbb{N}+1.$

Proof:

From Lemma 3.1 such that

$\frac{d(A_n)}{n}=h(R_n)(since|R_n|=n).$

Then $d(A_n)=nh(R_n)=(2m+1)h(R_n)$ where $n=2m+1\mathrm{\forall }m\in \mathbb{N}$

Thus, by Theorem 3.1 and Remark 3.3

$d(A_n)=(2\rho +1)h(R_n).$

Remark 3.4. Theorem 3.2 represents the trace of natural rhotrix of dimension n. This is unique with the natural rhotrix only.

Corollary 3.1. Let $R_n$ be a natural rhotrix, and $A_n=\{a_1,a_2,\dots ,a_n\}$ be the set of ordered elements along any of the diagonals of $R_n$, then $d(A_n)=n(m^2+m+1)\mathrm{\forall }m\in \mathbb{N}$ and $n=2m+1$.

Proof:

Since $h(R_n)=m^2+m+1\mathrm{\forall }m\in \mathbb{N}$ and $n=2m+1$. The proof follows from Theorem 3.2.

Remark 3.5. (i) Corollary 3.1 is used to determine the sum of elements in any of the diagonals of an arbitrary natural rhotrix. For example, the sum of elements of any of the diagonals of a natural rhotrix of dimensions: $3$ is $9$, $5$ is $35$ and $7$ is $91$ e.t.c. (ii) This also corresponds to the trace of $R_n$ which is given as $tr(R_n)=n(m^2+m+1)\mathrm{\forall }m\in \mathbb{N}$ and $n=2m+1$.

Theorem 3.3. Let $R_n$ be a natural rhotrix, and $A_n=\{a_1,a_2,\dots ,a_n\}$ be the set of elements in the diagonal of $R_n$. Then $d(A_n)=2codet(R_n)+det(R_n)$.

Proof:

Since

$2codet(R_n)=\rho det(R_n)=\rho h(R_n)$

then

$2codet(R_n)=2\rho h(R_n).$

But

$d(A_n)=(2\rho +1)h(R_n)=2\rho h(R_n)+h(R_n).$

Thus

$d(A_n)=2codet(R_n)+det(R_n).$

Remark 3.6. Invariably, $d(B_n)=2codet(R_n)+det(R_n)$ where $B_n=\{b_1,b_2,\dots ,b_n\}$ the set of entries along the minor diagonal of $R_n$, which gives the same value.

Theorem 3.4. Given any natural rhotrix $R_n$, and $d(A_n)$ the sum of elements in any of the diagonals of $R_n$. Then $d(A_n)=\rho (|R_n|+1)+h(R_n)\mathrm{\forall }n\in 2\mathbb{N}+1$, where $h(R_n)$, $\rho$ and $|R_n|$ are the heart, index and cardinality of $R_n$ respectively.

Proof:

Since $codet(R_n)=\frac{\rho }{2}(|R_n|+1)$(Theorem 2.3) then the proof follows from Theorem 3.3.

Corollary 3.2 Given any natural rhotrix $R_n$, then $d(A_n)=\frac{2\rho +1}{2}(|R_n|+1)$

Proof:

Since

$codet(R_n)=\frac{\rho }{2}(|R_n|+1)$

then,

$det(R_n)=\frac{1}{2}(|R_n|+1)$

Thus, the proof follows from Theorem 3.3.

Theorem 3.5. Let $R_n$ be a natural rhotrix and $U_n=\{e_1,e_2,\dots ,e_t\}$ be the set of all ordered elements or entries in $R_n$, then the sum of all elements is simply given as:

$\sum _{i=1}^t{e}_i=h(R_n)|R_n|$

where $h(R_n)$ and $|R_n|$ are the heart and the cardinality of $R_n$ respectively.

Proof:

Since $e_1,e_2,\dots ,e_t$ represent all the entries, in a specified order of $R_n$, for all non-zero $e_i\in \mathbb{N}$, then

$\sum _{i=1}^t{e}_i=\frac{t}{2}[2a+(t-1)d]$

Now, for all $t=|R_n|,n\in 2\mathbb{N}+1$ and $m\in \mathbb{N}$, we have

$\sum _{i=1}^t{e}_i=2m^4+4m^3+5m^2+3m+1$

$=(2m^2+2m+1)(m^2+m+1)$

$=h(R_n)|R_n|.$

Corollary 3.3. Let $R_n$ be a natural rhotrix and $\{e_1,e_2,\dots ,e_t\}$ be the set of all entries of $R_n$, starting from the first non-zero element, then

$\sum _{i=1}^t{e}_i=th(R_n)t\in 2\mathbb{N}+1$

Proof:

Since $R_n$ is a well-ordered natural rhotrix, then $t=|R_n|$. The proof follows.

Corollary 3.4. Let $R_n$ be a natural rhotrix and $\{e_1,e_2,\dots ,e_t\}$ the entries of $R_n$, then

$\sum _{i=1}^t{e}_i=|R_n|det(R_n)$

Proof:

Since $det(R_n)=h(R_n)$, then the proof follows.

Corollary 3.5. Let $R_n$ be a natural rhotrix and $\{e_1,e_2,\dots ,e_t\}$ the set of all entries of $R_n$, then

$codet(R_n)=\frac{\rho }{|R_n|}\sum _{i=1}^t{e}_i$

Proof:

Recall

$codet(R_n)=\rho det(R_n)$

then,

$det(R_n)=\frac{1}{\rho }codet(R_n)$

Thus, from Theorem 3.5

$codet(R_n)=\frac{\rho }{|R_n|}\sum _{i=1}^t{e}_i.$

Corollary 3.6. Let $R_n$ be a natural rhotrix and $\{e_1,e_2,\dots ,e_t\}$ the set of all entries of $R_n$, then

$codet(R_n)=\frac{m}{|R_n|}\sum _{i=1}^t{e}_i\mathrm{\forall }m\in \mathbb{N},n=2m+1$

Proof:

The proof follows from Corollary 3.5.

Theorem 3.6. Let $R_n$ be $n$-dimensional natural rhotrix and $\{e_1,e_2,\dots ,e_t\}$ be the set of all entries in $R_n$. If $A_n=\{a_1,a_2,\dots ,a_n\}$ is the set of all entries along the major or minor diagonal of $R_n$. Then,

$d(A_n)=\frac{2\rho +1}{|R_n|}\sum _{i=1}^t{e}_i,n\in 2\mathbb{N}+1$

Proof:

$d(A_n)=2codet(R_n)+det(R_n)$

$=2\frac{\rho }{|R_n|}\sum _{i=1}^t{e}_i+\frac{1}{|R_n|}\sum _{i=1}^t{e}_i$

$=\frac{2\rho +1}{|R_n|}\sum _{i=1}^t{e}_i$

Corollary 3.7. Let $R_n$ be a natural rhotrix and $\{e_1,e_2,\dots ,e_t\}$ be the set of all entries of $R_n$ starting from the first non-zero element of $\mathbb{N}$, then

$d(A_n)=\frac{n}{t}\sum _{i=1}^t{e}_i\mathrm{\forall }n\in \mathbb{N}$

Proof:

Since $2\rho +1=n,\rho \in \mathbb{N}$. Then, the proof follows.

Theorem 3.7. Let $R_n$ be a natural rhotrix, and $d(A_n)$ the sum of elements along the major or minor diagonal of $R_n$. If $d(A_n^c)$ denotes the sum of the elements outside the diagonals of $R_n$. Then,

$d(A_n^c)=(|R_n|+1-2n)h(R_n)$

Proof:

Since

$\sum _{i=1}^t{e}_i=2d(A_n)+d(A_n^c)-h(R_n)$

$d(A_n^c)=\sum _{i=t}^t{e}_i-2d(A_n)+h(R_n)$

$=|R_n|h(R_n)-2nh(R_n)+h(R_n)$

$=(|R_n|+1-2n)h(R_n)n\in 2\mathbb{N}+1$

Remark 3.7. If $|R_n|=t$ in case of well-ordered natural number where the first non-zero element is $1$ then

$d(A_n^c)=(t+1-2n)h(R_n)$

4. Conclusion

Rhotrix algebra has found applications in mathematical sciences, theoretical physics, engineering, architecture and computer science-particularly in cryptography. It has been observed that architects and engineers now design structures more in rhomboidal shapes rather than the traditional rectangular pattern for aesthetic purposes. Therefore, this work provided an in-depth scientific study of the rhomboidal structure called the natural rhotrix. It examined further properties of the rhotrix. The concept of diagonal function was introduced. Several mathematical links between the heart and other functions of the rhotrix set were established and presented. All these were put together as theory of this beautiful rhotrix. The several mathematical links testify to its beauty. Equations are everywhere, and more of it are yet to be discovered. However, if the elements are not ordered, then it becomes a challenge. This is an area for future work.

Acknowledgements

The author wishes to thank all the anonymous reviewers for their contributions.

Additional information

Funding

The author received no direct funding for this research.

Notes on contributors

A. O. Isere

A. O. Isere obtained first and second degrees in Mathematics from the University of Benin, Benin City, Nigeria, in 1999 and 2004, respectively. He joined Ambrose Alli University, Ekpoma, Nigeria in 2006, as a lecturer and researcher in Mathematics till date. Between 2008 and 2014, he published some articles on the outbreaks and control of cholera in Nigeria. He got his Ph.D in Mathematics(Algebra) from Federal University of Agriculture, Abeokuta, Nigeria, in 2014. He specializes in loop theory, precisely in Osborn loops. Since then, he has published various articles on finite Osborn loops, Holomorphy of Osborn loops and on characterization of Non-Universal Osborn loops of order 4n. In 2015, he picked interest in rhotrix algebra, and has published several articles, which includes natural rhotrix, rhotrix quasigroups and rhotrix loops inter alia. In 2018, he introduced even-dimensional rhotrix. His research projects also include quandles and their applications.