CHARACTERIZATION OF A HEART-ORIENTED PARALETRIX

This paper presents more results in the theory of paraletrix. These results are simply a characterization of a heart-oriented paraletrix ring, which include paraletrix integral domain, paraletrix polynomial, paraletrix ring functions, differentiation and integration.


INTRODUCTION
In [4], Atanassov and Shanaon discussed arrays of numbers that are in some way, between twodimensional vectors and (2 × 2)-dimensional matrices in their paper titled matrix-tertions and noitrets. As an extension, Ajibade [1] in 2003 introduced objects which are in some ways, between (2 × 2) -dimensional and (3 × 3) -dimensional matrices. This field of science now known as rhotrix theory was defined in [1] for dimension three as: where = ℎ( ) is called the heart of any rhotrix and ℝ is the set of real numbers.
It is worthy to note that these heart-oriented rhotrices are always of odd dimension. Thereafter, Mohammed [10] in his PhD thesis extended the idea to rhotrix set of size .
It is known in [1] that addition and multiplication of two heart-oriented rhotrices are as follows: respectively. Furthermore, Mohammed [10] and Ezegwu et al [5] gave a generalization of this heart-oriented rhotrices.
A row-column multiplication of heart-oriented rhotrices was given by Sani [15] as: Sani [16] also gave a generalization of this row-column multiplication of heart-oriented rhotrices as: where and denote -dimensional rhotrices (with rows and columns).
Mohammed [9] classified the heart-oriented rhotrices as abstract structures of rings, fields, integral domains and unique factorization domain. The rhotrix quadratic polynomial presented as part of a note on rhotrix exponent rule and its applications in [10] was extended in [18]. Rhotrix polynomial and its extension to construction of rhotrix polynomial ring was studied in [19]. Also in [11], some construction of rhotrix semigroup was given and then extended by in [13] to rhotrix type A semigroup. The study of non-commutative full rhotrix ring and its subring was 3132 R. U. NDUBUISI, R. B. ABUBAKAR, O. G. UDOAKA, I. J. UGBENE carried out by Mohammed in [12]. Patil [14] gave a characterization of ideals of rhotricesover a ring and its application.
Consequently, Isere [6,7] introduced rhotrices without a heart, and these rhotrices were found to be even-dimensional rhotrices.
Tudunkaya and Makanjuola [17] studied rhotrices and the construction of finite fields.
The concept of paraletrix was introduced by Aminu and Michael [2] as an extension of rhotrix [1] when the number of rows is not equal to the number of columns. It is worthy to note that not all paraletrix has a heart as seen in [2].

CHARACTERIZATION OF A HEART-ORIENTED PARALETRIX
It is important to note that the multiplication of paraletrix and ′ using Sani [15] is only possible whenever the number of columns of is equal to the number of rows of ′ for any arbitrary paraletrix.
The objective of this paper is to classify paraletrices as rings using Ajibade [1]. The result in this paper is an extension to the ones given in [9].

PRELIMINARIES
In this section we recall some definitions as well as some known results which will be useful in this paper. For notation and terminologies not mentioned in this paper, the reader is referred to [1], [15], [16], [7], [12] and [2] respectively.
Throughout this paper, we will use to denote any paraletrix, while and are the number of rows and columns of an arbitrary paraletrix, where , (2 + 1 ∶ ℕ). . For more information, the reader is referred to [2].
Definition 2.2. The cardinality or order of a paraletrix is defined to be the number of entries of an arbitrary paraletrix with number of rows and number of columns. This is denoted by Lemma 2.3 [2]. Let be an × -dimensional paraletrix. If has a heart then the heart is unique.
Theorem 2.4 [2]. A necessary and sufficient condition for the heart of an × -dimensional paraletrix to exist is that the order is odd.
Theorem 2.5 [2]. A necessary and sufficient condition for the heart of an × -dimensional paraletrix to exist is that | − | = 4 where = 0,1,2, … Definition 2.6. Suppose and are two × -dimensional and × -dimensional heartoriented paraletrices such that . = then is said to be the identity paraletrix. The identity of a paraletrix with 3 rows and 7 columns is given by A ring is defined as a non-empty set with two compositions +,∘ ∶ × → with the following properties; iii) for all , , the distributivity laws are satisfied; It is worthy to note that a ring such that for , , = is referred to as a regular ring.

Definition 2.8. A subgroup of ( , +) is said to be a left ideal if ⊂ , and a right ideal if
if ⊂ . is said to be an ideal if it is both a left and right ideal. is a subring if ⊂ . It is worthy to note that every left or right ideal in is also a subring of . The intersection of (arbitrary many) (left, right) ideals is again a (left, right) ideal. Definition 2.9. Let , be rings. A mapping ∶ → is called a ring morphism or Definition 2.10. Let be a ring, we say that is a differential ring if it is equipped with one or more derivations, that are homomorphisms of additive groups ∶ → such that each derivation satisfies the Leibniz product rule ( 1 2 ) = ( 1 ) 2 + 1 ( 2 ) for every 1 , 2 .

RING OF PARALETRICES
In this section, we apply the notions of rings in the development of new abstract structure of paraletrices with respect to the binary operations of addition (+) and multiplication (∘). The paraletrix under consideration will be a 3 × 7-dimensional heart-oriented paraletrix. Now let * = 〈 , +, ∘〉 be an abstract structure consisting of the set of all real a 3 × 7 paraletrices together with the operations of addition (+) and multiplication (∘). Let the identity of be as in Definition 2.5 while the zero paraletrix be such that the elements of the paraletrix are all zero.
We have the following results: Proof. It is obvious that * is an abelian group.
It is important to note that * is a commutative ring with identity denoted by * . In * , it is obvious that the multiplication of nonzero paraletrices and is equal to zero. Hence * has zero divisors and is not an integral domain. Proof. That is an ideal of * implies that is a subset of * and ≠ ∅.
In [8], polynomial equations are defined over variables and coefficients which are also rhotrices.
In [2], it is known that we can extract 2 × 4 and 1 × 3 dimensional matrices from a 3 × 7dimensional paraletrix. It is also known in [11] that a rhotrix semigroup can be embedded in a matrix semigroup. The type A version of the embedding was proved in [13]. [12] proved that a rhotrix ring can be embedded in a matrix ring. which is undefined. Thus ( ∘ ′ ) ≠ . ′ so that ∶ * = 〈 , +, ∘〉 → = ( × , +, . ) is not a ring morphism.

DIFFERENTIATION OF *
In [3], differentiation and integration of rhotrices was presented. In this section, we will prove that analogous result holds for ring of paraletrices.
Let With this we have the following results.  Proof. It follows from the multiplication of two paraletrices as defined by Ajibade [1].  Remark 4.4. From Lemma 4.2, it is clear that the ring * is a differential ring.

REPRESENTATION
In this section, we shall show that an arbitrary ring can be represented as a 3 × 7-dimensional heart-oriented paraletrix. In particular, it will be shown that an arbitrary ring of numbers is isomorphic to a 3 × 7-dimensional heart-oriented paraletrix. This result can be considered as an analogous case of Cayley's theorem in ring theorem. It is obvious that the map is well defined. It is also one-to-one since for ℎ 1 , ℎ 2 , ℎ 1 = ℎ 2 which implies that ℎ 1 = ℎ 2 . That is a homomorphism follows from the fact that where ℎ ′ = ℎ 1 + ℎ 2 , ℎ ′′ = ℎ 1 . ℎ 2 .
Furthermore, we have that That is, each element in has an image in ( ). So the image of under is the whole of the heart-oriented paraletrix ring. Thus, = ( ). Hence, is an isomorphism for any arbitrary ring to the heart-oriented paraletrix ring * . The proof is then complete.

SUMMARY
This paper considered classification of a paraletrix as an abstract structure as a follow up of known results on rhotrices.
This work showed that the results in rhotrix theory should necessarily hold in paraletrix theory.
However, it is scholastic to examine further results of the heart-oriented paraletrices vis a vis the heart-oriented rhotices. These results will contribute greatly in paraletrix algebra. The introduction of a heartless paraletrix will be presented in another paper.