Hindawi Computational and Mathematical Methods in Medicine Volume 2017, Article ID 1975780, 6 pages https://doi.org/10.1155/2017/1975780

Research Article Mathematical Modeling for Inherited Diseases

Saima Anis,1 Madad Khan,1 and Saqib Khan2

1 Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad, Pakistan 2Department of Mathematics, Government Postgraduate College, Abbottabad, Pakistan

Correspondence should be addressed to Madad Khan; [email protected]

Received 22 December 2016; Revised 7 April 2017; Accepted 24 May 2017; Published 9 July 2017

Academic Editor: Delfim F. M. Torres

Copyright © 2017 Saima Anis et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We introduced a new nonassociative algebra, namely, left almost algebra, and discussed some of its genetic properties. We discussed the relation of this algebra with flexible algebra, Jordan algebra, and generalized Jordan algebra.

1. Introduction 23 pairs of chromosomes, with a total of 46 chromosomes. A gene is a unit of hereditary information and lies on chro- We introduced a new nonassociative and noncommutative mosomes. A gene can take different forms which are called algebra which has several properties similar to nonassociative alleles. In these 23 pairs, one represents the sexual character andcommutativealgebras.Therelationoftheleftalmost in males; a locus which occurs on 22 pairs of chromosomes algebra with other nonassociative algebras is useful and inter- is called autosomal, whereas a locus on one pair is called esting to be known; in this regard, we found some relations sex-linked. Diploid cells are those which carry a double set with known algebras, namely, flexible, alternative, and Jordan of chromosomes and haploid cells are those which carry a algebras. We discussed some characteristics of this algebra single set of chromosomes. Diploid cells produce sex cells which is similar to a commutative and associative algebra. We called gametes through a process called meiosis. Mitosis is discussed some of the genetic properties of this algebra. the process of reproduction in haploid cells. The gametes cells meet each other and give a zygote which is again a diploid cell. Some Basics from Genetics. Here, we discuss some simple ideas in genetics for nongeneticists and for those who are Basics of Genetic Algebra. After the theory of Charles Darwin, purely related to mathematics. Each cell of an organism it was Gregor Mendel who studied the natural laws of genetic contains long thread-like structures called chromosomes inheritance and tried to express them in a mathematical which are located inside the nucleus of animals and plants. language. In [1–3], Etherington introduced a new method Chromosomes are made of protein and a single molecule of nonassociative algebras to study genetics. Holgate stud- of deoxyribonucleic acid (DNA). After cell division, chro- ied these algebras with genetic realizations in [4–7]. Reed mosomes pass from parent cells to the newborn cells. The discussed the nonassociative algebraic structure of genetic particular intersections that the DNA carries make each inheritance in [8]. The genetic algebra with mutation has living creature unique from others. The molecules of the DNA been discussed in [9, 10]. Several other authors studied aretoolong,whichcanbefittedinsidethecellsonlyby nonassociative algebras with genetic realizations (for details, chromosomes. Moreover, chromosomes play an important see [10–14]). role in copying and distributing DNA accurately in the whole Let 𝑅 be a nonempty set together with two binary process of cell division. Problems in chromosomes in new operations “+” and “⋅”whichsatisfiesalltheaxiomsofan cells can create serious issues like leukemia and some types of associative ring (algebra) except an associative property with cancer. Males and females have different chromosomes; that respecttomultiplication;then,itisknownasanonassociative is, females have two 𝑋 chromosomes in their cells whereas ring (algebra). Lie ring was introduced by defining and males have one 𝑋 and one 𝑌 chromosome. Humans have replacing a new binary operation [𝑎, 𝑏] = 𝑎𝑏,forall −𝑏𝑎 𝑎,, 𝑏 2 Computational and Mathematical Methods in Medicine

with ordinary multiplication of an associative ring (algebra); the juxtaposition between 𝑥1 and 𝑥2 shows a binary operation obviously, it is a nonassociative ring (algebra). By defining a which is not associative because (𝑥1𝑥2)𝑥2 =𝑥̸ 1(𝑥2𝑥2) or new binary operation 𝑎 ⋅ 𝑏 = (1/2)(𝑎𝑏 +𝑏𝑎) on an associative equivalently (𝑥1𝑥2)𝑥2 =𝑥1(𝑥2𝑥2) says that both alleles algebra over a field whose characteristic is not equal to 2, we 𝑥1 and 𝑥2 are the same. Therefore, the associative law obtain another important nonassociative algebra known as does not hold for the gametic algebra and it also does not Jordan algebra. It is worth mentioning here that the theory hold for the zygotic algebra. Generally, algebras associated of nonassociative algebras is a fruitful branch of algebra. with genetics are commutative but nonassociative. In [3], Most importantly, the class of nonassociative algebras has Etherington proved that the zygotic algebra can be obtained closed connections with other branches of mathematics; from the gametic algebra through a commutative duplication also, it has closed connections with quantum mechanics, process. Now, it is interesting to note that there is a class of physics, biology, and other sciences. The crucial part of this algebras which is nonassociative and noncommutative but theory is the theory of nearly associative rings and algebras: possesses many characteristics similar to commutative and Lie, alternative, and Jordan algebras. In short, considering associativealgebrasandhascloserelationswithcommutative nonassociative algebras over real number fields has several algebras. Using notions from these algebras, we give a more applications in biology especially in genetics. Moreover, there general definition of gametic and zygotic algebras. Moreover, are some other classes of nonassociative algebras closely this nonassociative and noncommutative algebra has closed related to genetics which are popular among mathematicians connections with Jordan and flexible algebras. In addition and geneticists for their usefulness in genetics. Generally, to this, if it contains a left identity, then it satisfies the thesetypesofalgebrasarecommutativeandnonassociative. famous Jordan identity and the generalized Jordan identity. In fact, one can study the properties of genetics by making This algebra works mostly like a commutative algebra; for 2 2 2 2 the mathematical models using nonassociative, commutative instance, 𝑥 𝑦 =𝑦𝑥 ,forall𝑥, 𝑦,holdsinacommutative algebras.Tovisualizetheconceptofsuchalgebras,letuspay algebra while this equation also holds for a left almost attention to some specific classes of algebras like gametic, algebra, and if a left almost algebra contains a left identity zygotic, and copular algebras. 𝑒,then𝑥𝑦 = (𝑦𝑥)𝑒,forany𝑥,.Infact,thestructureis 𝑦 Here, we will discuss some simple algebras with genetic nonassociative and noncommutative but it possesses many realizations. In order to understand the algebraic properties properties which usually hold in associative and commutative in genetics, simple Mendelian inheritance has been consid- algebraic structures. Also, defining a new operation on this ered in [10]. If we consider a single gene with two alleles 𝑥1 algebra gives a commutative and associative algebra. and 𝑥2,aftermatingofalleles𝑥1 and 𝑥2, we get the zygotes In this paper, we will discuss those algebras which are 𝑥1𝑥1, 𝑥1𝑥2, 𝑥2𝑥1,and𝑥2𝑥2,where𝑥1𝑥1,𝑥2𝑥2 are known as nonassociative and noncommutative but have close connec- homozygotes while 𝑥1𝑥2,𝑥2𝑥1 are called heterozygotes. In tions with nonassociative and commutative algebras. We particular, 𝑥1𝑥2 =𝑥2𝑥1,whichmeansthatthecommutative generalized the definition available in [8] and introduced a law holds in this case, which genetically represents the notion new generalized definition for gamete algebras. Moreover, we that mating of allele 𝑥1 with allele 𝑥2 is the same as mating of introduce a new class of a nonassociative algebra called left 𝑥2 with 𝑥1. Let us consider the equation 𝑥1𝑥2 = (1/2)(𝑥1 + almost algebra. 𝑥2). This linear equation represents the notion that 𝑥1𝑥1 = Werestrictourselvesbyconsideringageneonaparticular 𝑥1 and 𝑥2𝑥2 =𝑥2. The algebra with the multiplication locus on a chromosome. Here, we begin with the definition tableaboveisknownasgametealgebra.Moreover,the of gametic algebra [8]. Consider 𝑛 gametes 𝑎1,𝑎2,...,𝑎𝑛 zygotic algebra can be obtained from the gametic algebra by asbasiselementsofan𝑛-dimensional real vector space. commutative duplication. From mathematical perspectives, Multiplication is defined by

𝑛 𝑎𝑖𝑎𝑗 = ∑𝛾𝑖𝑗𝑘 𝑎𝑘, 𝑘=1 𝑛 (1) such that 0≤𝛾𝑖𝑗𝑘 ≤1, 𝑖,𝑗,𝑘=1,2,...,𝑛, ∑𝛾𝑖𝑗𝑘 =1, 𝑖,𝑗,𝑘=1,2,...,𝑛,𝑖𝑗𝑘 𝛾 =𝛾𝑗𝑖𝑘, 𝑖,𝑗,𝑘=1,2,...𝑛, 𝑘=1

where 𝛾𝑖𝑗𝑘 represent the relative gene frequencies. Let 𝑎, 𝑏,,where 𝑐∈𝑆 𝑆 is a left almost semigroup with left The resulting algebra S is called an 𝑛-dimensional identity 𝑒.Then, gametic algebra. 𝑎 (𝑏𝑐) =𝑏(𝑎𝑐) , ∀𝑎, 𝑏, 𝑐 ∈𝑆. (2) Some Basic Notions of Nonassociative Algebras.Agroupoid𝑆 is called a left almost semigroup if it satisfies the following left But the converse of the above statement is not true. invertive law, (𝑎𝑏)𝑐 = ,forall(𝑐𝑏)𝑎 𝑎, 𝑏,.Holgatecalled 𝑐∈𝑆 If a left almost semigroup contains a right identity, then it italeftinvertivegroupoid[4].Anelement𝑒 of a groupoid 𝑆 becomes a commutative semigroup. A left almost semigroup is called left (right) identity if 𝑒𝑥=𝑥(𝑥𝑒=𝑥)for all 𝑥 in 𝑆.A 𝑆 is a mid structure between a groupoid and a commutative left identity in a left almost semigroup is unique. semigroup. Computational and Mathematical Methods in Medicine 3

From the discussion above, it is easy to conclude that where 0≤𝛾𝑖𝑗𝑘 ≤1for all 𝑖,𝑗,𝑘 = 1,2,...,𝑛, 𝛾𝑖𝑗𝑘 represent this nonassociative structure with left identity has a closed the relative gene frequencies. connection with a commutative semigroup. Then, the resulting algebra A is called an 𝑛-dimensional AnonassociativealgebraA isavectorspaceoverafield nonassociative and noncommutative left almost gametic F along with the bilinear multiplication from A × A → A, algebra. satisfying the following distributive properties: Let (𝑎1,...,𝑎𝑛) be a basis representing the 𝑛 alleles gen- erating a gametic left almost algebra and the multiplication (𝛼𝑎 + 𝛽𝑏) 𝑐=𝛼 (𝑎𝑐) +𝛽(𝑏𝑐) , defined as 𝑎𝑖𝑎𝑗 = (1/2)(𝑎𝑖 +𝑎𝑗). Consider the mapping 𝜔: V → R and let 𝑤 be the weight function defined as 𝜔(𝑎𝑖)=1. 𝑎(𝛼𝑏+𝛽𝑐)=𝛼(𝑎𝑏) +𝛽(𝑎𝑐) , (3) For any element 𝑥 of A, 𝑥=∑𝛼𝑖𝑎𝑖.Thus, ∀𝛼, 𝛽 ∈ F,𝑎,𝑏,𝑐∈A.

A 𝑛 𝑛 𝑛 𝑛 Anonassociativealgebra is called left almost algebra 𝑥2 =𝑥𝑥=∑ (𝛼 𝑎 ) ∑ (𝛼 𝑎 )=∑ ∑ (𝛼 𝑎 )(𝛼𝑎 ) F 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 over a field if it satisfies the left invertive property with 𝑖=1 𝑖=1 𝑖=1 𝑖=1 respect to multiplication. 𝑛 𝑛 𝑛 𝑛 Several authors discussed mostly commutative and = ∑ ∑ (𝛼 𝛼 )(𝑎𝑎 )=∑ ∑ (𝛼 𝛼 )𝑎 nonassociative algebras with genetic realizations. There are 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖=1 𝑖=1 𝑖=1 𝑖=1 some cases of noncommutative, nonassociative algebraic (5) structures which were discussed in [2, 10, 14]. Moreover, 𝑛 𝑛 𝑛 𝑛 = (𝛼 𝛼 )(𝑎)= (𝛼 ) (𝛼 𝑎 )=𝑥, Mendel’s algebra is interesting to discuss. To study such ∑ ∑ 𝑖 𝑖 𝑖 ∑ 𝑖 ∑ 𝑖 𝑖 cases of Mendel’s algebra with mutation, we introduce a 𝑖=1 𝑖=1 𝑖=1 𝑖=1 new class of nonassociative and noncommutative algebras 𝑛 known as left almost algebras. These algebras possess many Provided ∑ (𝛼𝑖)=1. characteristics which are similar to those of commutative 𝑖=1 nonassociative algebras with genetic realizations. Here, we give the generalized definition for gametic algebra; consider 𝑛 gametes 𝑎1,𝑎2,...,𝑎𝑛 asbasiselementsofan𝑛-dimensional Now, in a similar way, we can define the zygotic algebra 𝑛 𝑎 =𝑎𝑎 left almost vector space over R. If we define multiplication by and for this we consider gametes 𝑖𝑗 𝑖 𝑗 (considering only 𝑖≤𝑗), and then random mating of zygotes 𝑎𝑖𝑗 with 𝑎𝑖𝑎𝑗 =𝛾𝑖𝑗1 𝑎1 +𝛾𝑖𝑗2 𝑎2 +𝛾𝑖𝑗3 𝑎3 +⋅⋅⋅+𝛾𝑖𝑗𝑛 𝑎𝑛, 𝑎𝑝𝑞 will produce zygotes 𝑎𝑘𝑠 with a particular ratio, say (4) 𝛾𝑖𝑗,𝑝𝑞,𝑘𝑠. Thus, we define the following generalized zygotic 1=𝛾 +𝛾 +𝛾 +⋅⋅⋅+𝛾 , 𝑖,𝑗,𝑘=1,2,...,𝑛, such that 𝑖𝑗1 𝑖𝑗2 𝑖𝑗3 𝑖𝑗𝑛 multiplication as

𝑎 𝑎 = ∑𝛾 𝑎 , ∑𝛾 =1, 𝑖,𝑗,𝑘=1,2,...,𝑛 0≤𝛾 ≤1 ∀𝑖,𝑗,𝑘=1,2,...,𝑛, 𝑖𝑗 𝑝𝑞 𝑖𝑗,𝑝𝑞,𝑘𝑠 𝑘𝑠 such that 𝑖𝑗,𝑝𝑞,𝑘𝑠 such that 𝑖𝑗,𝑝𝑞,𝑘𝑠 (6) 𝑘≤𝑠 𝑘≤𝑠

where 𝛾𝑖𝑗,𝑝𝑞,𝑘𝑠 are the relative gene frequencies. 2. Mutation Algebra An element 𝑥 in our noncommutative, nonassociative algebra A indicates a population or a gene pool and it can In [9], the author considered mutation algebra with mutation 𝑟 𝑠 be expressed as a linear combination of the basis elements rates and , the gametic algebra having the basis with 𝐷 𝑅 𝑎1,𝑎2,...,𝑎𝑛 as 𝑥=𝜆1𝑎1 +𝜆2𝑎2 +𝜆3𝑎3 + ⋅⋅⋅ + 𝜆𝑛𝑎𝑛 and elements and , where the multiplication table is defined 𝜆1 +𝜆2 +𝜆3 +⋅⋅⋅+𝜆𝑛 =1,where𝜆𝑙 ∈ F,forall𝑙=1,2,...,𝑛. as The algebra with genetic realization arising from expres- 𝐷2 = (1−𝑟) 𝐷+𝑟𝑅, sion (4) is more general than the one arising from (1) because (9) 2 the algebra is both nonassociative and noncommutative. 𝑅 = s𝐷+(1−𝑠) 𝑅 An algebra A iscalledflexibleifitsatisfiesthefollowing 1 1 property: 𝐷𝑅 = (1−𝑟+𝑠) 𝐷+ (1−𝑠+𝑟) 𝑅. 2 2 (10) (𝑥𝑦) 𝑥 = 𝑥 (𝑦𝑥) ,∀𝑥,𝑦 in A. (7) Then, the author chose another basis with elements 𝑎=𝐷 A An algebra is called a generalized Jordan algebra if it and 𝑏=𝐷−𝑅,andthus satisfies the following property: 𝑎2 = 𝑎 − 𝑟𝑏, (𝑥𝑦) (𝑥𝑥) =𝑥(𝑦 (𝑥𝑥)) ,∀𝑥,𝑦in A. (8) 1 It is obvious that both flexible and generalized Jordan 𝑎𝑏 = (1−𝑟−𝑠) 𝑏, (11) 2 2 algebras are different but if 𝑥=𝑥 for all 𝑥 then both become 2 identical. 𝑏 =0. 4 Computational and Mathematical Methods in Medicine

Let us define an abstract algebra A generated by {𝑎𝑖 :1≤ Proof. Clearly, M4(tF) is a left almost algebra. Next, we will 𝑖≤𝑛}over a finite field F. If we define the binary operation prove that it satisfies the property of flexible algebra. We have 2 2 2 2 “⋆”onA as 𝑎𝑖 ⋆𝑎𝑗 =𝛼𝑎𝑖 +𝛼𝑎𝑗(𝛼+𝛼 =1),thenthis 𝑎𝑗 ⋆𝑎𝑗 =𝛼𝑎𝑗 +𝛼 𝑎𝑗 =(𝛼+𝛼)𝑎𝑗 =𝑎𝑗. Therefore, algebra satisfies (10) but 𝑎𝑖 ⋆𝑎𝑖 =𝑎𝑖. Therefore, it is important to mention here that the algebra defined by this operation is 𝑛 𝑛 𝑛 𝑛 𝑋⋆𝑋=∑𝛽 𝑎 ⋆ ∑𝛽 𝑎 = ∑ ∑𝛽 𝛽 (𝑎 ⋆𝑎) not totally consistent with the mutation algebra introduced 𝑗 𝑗 𝑗 𝑗 𝑗 𝑗 𝑗 𝑗 by Gonshor above. This simply implies that there is a lack of 𝑗=1 𝑗=1 𝑗=1 𝑗=1 (15) one hundred correspondences between this algebra and the 𝑛 𝑛 𝑛 𝑛 𝑛 algebra defined in (10) but there are still several similarities = ∑ ∑𝛽𝑗𝛽𝑗𝑎𝑗 = ∑𝛽𝑗∑𝛽𝑗𝑎𝑗 = ∑𝛽𝑗𝑎𝑗. existing between the ideas of the mutation algebra and the 𝑗=1 𝑗=1 𝑗=1 𝑗=1 𝑗=1 algebraweintroduced.Inthenexttheorem,wewillprovethat this algebra is a left almost algebra. We denote this algebra by Thus, Mn(𝛼F). (𝑋⋆𝑌) ⋆ (𝑋⋆𝑋) = [(𝑋⋆𝑋) ⋆𝑌] ⋆𝑋 Theorem 1. Mn(𝛼F) is a noncommutative and nonassociative = [(𝑌⋆𝑋) ⋆𝑋] ⋆𝑋 left almost algebra. = (𝑋⋆𝑋) ⋆ (𝑌⋆𝑋) (16) Proof. Obviously, Mn(𝛼F) is closed. Next, we will show that Mn(𝛼F) satisfiestheleftinvertiveproperty,forthisleft =𝑋⋆(𝑌⋆𝑋) 𝑛 𝑛 𝑋, 𝑌, 𝑍∈ Mn(𝛼F).Then,𝑋=∑𝑗=1 𝛽𝑗𝑎𝑗, 𝑌=∑𝑘=1 𝛾𝑘𝑎𝑘 and 𝑛 =𝑋⋆[𝑌⋆(𝑋⋆𝑋)] . 𝑍=∑𝑙=1 𝛿𝑙𝑎𝑙.Toprove(𝑋 ⋆ 𝑌) ⋆ 𝑍 = (𝑍, ⋆𝑌)𝑋 we need to prove (𝑎𝑗 ⋆𝑎𝑘)⋆𝑎𝑙 =(𝑎𝑙 ⋆𝑎𝑘)⋆𝑎𝑗.

2 2 (𝑎𝑗 ⋆𝑎𝑘)⋆𝑎𝑙 =𝛼(𝛼𝑎𝑗 +𝛼 𝑎𝑘)+𝛼 𝑎𝑙 Corollary 3. The algebra M4(tF) is a flexible algebra. 2 3 2 =𝛼𝑎𝑗 +𝛼 𝑎𝑘 +𝛼 𝑎𝑙, It is obvious from Theorem 2 that M4(tF) contains idempotent elements and we know that idempotent elements (𝑎 ⋆𝑎 )⋆𝑎 =𝛼(𝛼𝑎 +𝛼2𝑎 )+𝛼2𝑎 𝑙 𝑘 𝑗 𝑙 𝑘 𝑗 (12) in nonassociative algebras have their own importance. Thus, 2 3 2 we arrived at the following remark. =𝛼𝑎𝑙 +𝛼 𝑎𝑘 +𝛼 𝑎𝑗 Remark 4. From the biological point of view, the idempotents =𝛼2𝑥+𝛼3𝑦+𝛼2𝑧. in the algebra M4(tF) have their own usefulness. Since this algebra has several characteristics similar to a nonassociative It is not associative because algebra arising in genetics, the idempotent elements of this 2 2 𝑎𝑗 ⋆(𝑎𝑘 ⋆𝑎𝑙)=𝛼𝑎𝑗 +𝛼 (𝛼𝑎𝑘 +𝛼 𝑎𝑙) algebra may be used for equilibria of a population described (13) by some nonassociative algebras with genetic realizations. 3 4 =𝛼𝑎𝑗 +𝛼 𝑎𝑘 +𝛼 𝑎𝑙. In the following, we will consider some other nonasso- 2 2 Moreover, 𝑎𝑗 ⋆𝑎𝑘 =𝛼𝑎𝑗 +𝛼 𝑎𝑘 and 𝑎𝑘 ⋆𝑎𝑗 =𝛼𝑎𝑘 +𝛼 𝑎𝑗. Thus, ciative algebras and will discuss their relations with the left 𝑋𝑌 =𝑌𝑋̸ ,forsome𝑋, 𝑌. almost algebra mathematically. An alternative algebra A is a nonassociative algebra If we consider this theorem for a finite field of cardinality 2 2 satisfying the following properties: 4 which is the extension of Z2,then𝑡 +𝑡 = 1gives 𝑡 +𝑡+1 = 2 0. More generally, 𝑥 +𝑥+1 = 0is the quadratic equation 𝑥(𝑥𝑦)=(𝑥𝑥) 𝑦, representing the irreducible polynomial in Z2 = {0, 1}.Thus, 2 2 2 (𝑦𝑥)𝑥=𝑦(𝑥𝑥) , 𝐺𝐹(2 ) = {0,1,𝑡,𝑡 },andthus𝐺𝐹(2 )\{0}=F \{0}=⟨𝑡: (17) 𝑡3 = 1⟩= {1,𝑡,𝑡2} M (t ) . We denote this algebra by 4 F .Roots ∀𝑥, 𝑦 A. 2 in of the equation 𝑥 +𝑥=1are (−1 + √5)/2 and (−1 − √5)/2. Thus, Lemma 5. If A is a left almost alternative algebra, then 2 2 2 𝑥(𝑥𝑦) = (𝑥𝑥)𝑦 = (𝑦𝑥)𝑥 =𝑦(𝑥𝑥) and 𝑥 𝑦=𝑦𝑥,forall𝑥 𝑎𝑘 ⋆𝑎𝑗 =𝛼𝑎𝑘 +𝛼 𝑎𝑗 and 𝑦 in A. (14) 2 −1 ± √5 (𝛼 + 𝛼 =1, where 𝛼= ). Proof. Let 𝑥 and 𝑦 belong to A;then,𝑥(𝑥𝑦) = (𝑥𝑥)𝑦 = 2 2 2 (𝑦𝑥)𝑥 = ,so𝑦(𝑥𝑥) 𝑥 𝑦=𝑦𝑥,forall𝑥 and 𝑦. 𝑎 ⋆𝑎 =𝛼𝑎 +𝛼2𝑎 =(𝛼+𝛼2)𝑎 =𝑎. Obviously, 𝑗 𝑗 𝑗 𝑗 𝑗 𝑗 2 2 It is proved above that 𝑥 𝑦=𝑦𝑥 for all 𝑥 and 𝑦 in A. 2 2 Theorem 2. The algebra M4(tF) is a generalized Jordan Thus, 𝑥 𝑥=𝑥𝑥 for 𝑥 in A. Therefore, we can define powers algebra. of an element in A. Computational and Mathematical Methods in Medicine 5

Lemma 6. If A is a left almost alternative algebra that contains This theorem represents a mathematical model showing 2 2 a left identity e,thenA becomes commutative and associative that mating of 𝑥 with 𝑦𝑧 is the same as mating of 𝑥 with 𝑧𝑦. with identity. Proposition 11. (𝑎𝑏)(𝑐𝑑) = (𝑑𝑏)(𝑐𝑎),forall =(𝑑𝑐)(𝑏𝑎) 2 2 Proof. Since 𝑥 𝑦=𝑦𝑥,forall𝑥 and 𝑦, therefore 𝑥=e𝑥= 𝑎, 𝑏, 𝑐,𝑑 in A. 2 2 e 𝑥=𝑥e =𝑥e.Then, Proof. The proof is easy. 𝑥𝑦 = (𝑥𝑦) e = (e𝑦) 𝑥=𝑦𝑥. (18) 𝑎 𝑆 𝑎1 =𝑎 𝑎𝑛+1 =𝑎𝑛𝑎 𝑛 It is easy to see that commutativity and the left invertive law For any in ,weput and ,where is a give associativity. positive integer. For the rest of the paper, by A,weshallmeantheleft Proposition 12. A has associative powers. almost alternative algebra satisfying (2). Proof. The proof is easy. Lemma 7. 𝑦𝑥𝑛 =𝑥𝑛𝑦 𝑥, 𝑦 A 𝑛≥2. ,forall in and for 𝑚 𝑛 𝑚+𝑛 Proposition 13. 𝑎 𝑎 =𝑎 ,forall𝑎∈A and positive 2 2 Proof. We already proved that 𝑥 𝑦=𝑦𝑥,forall𝑥, 𝑦 in A. integers 𝑚, 𝑛. Then, Proof. According to Proposition 12, the result is true for 𝑚⩾ 2 2 2 2 2 (𝑥 𝑦)𝑥=(𝑦𝑥)𝑥=(𝑥𝑦) 𝑥 = (𝑥𝑦) 𝑥 =𝑥 (𝑦𝑥) 1. Again, by Lemma 7, we obtain

2 3 =𝑦(𝑥𝑥) = 𝑦𝑥 , (19) 𝑎𝑚+1𝑎𝑛 =(𝑎𝑚𝑎)𝑛 𝑎 =(𝑎𝑎𝑚)𝑎𝑛 =(𝑎𝑚𝑎𝑛)𝑎=𝑎𝑚+𝑛𝑎 (24) (𝑦𝑥2)𝑥=(𝑥𝑥2)𝑦=𝑥3𝑦. =𝑎𝑚+𝑛+1.

3 3 𝑘 𝑘 𝑚 𝑛 𝑚+𝑛 Thus, 𝑦𝑥 =𝑥𝑦. Let us assume that let 𝑦𝑥 =𝑥𝑦,for𝑘≥3. Hence, 𝑎 𝑎 =𝑎 ∀𝑎 ∈ A. Then, 𝑚 𝑛 𝑚𝑛 Proposition 14. (𝑎 ) =𝑎 for all 𝑎∈𝐴and positive inte- (𝑥𝑘𝑦) 𝑥 = (𝑥𝑦)𝑘 𝑥 =𝑥𝑘 (𝑦𝑥) = 𝑦 (𝑥𝑘𝑥) = 𝑦𝑥𝑘+1, gers 𝑚,. 𝑛 (20) 𝑘 𝑘 𝑘+1 (𝑦𝑥 )𝑥=(𝑥𝑥 )𝑦=𝑥 𝑦. Proof. The result is true for 𝑛=1.Supposeitistruefor𝑛>1. Then, we obtain

𝑚 𝑛+1 𝑚 𝑛 𝑚 𝑚𝑛 𝑚 𝑚𝑛+𝑚 𝑚(𝑛+1) (𝑎 ) =(𝑎 ) 𝑎 =𝑎 𝑎 =𝑎 =𝑎 . (25) Theorem 8. Every A becomes a generalized Jordan algebra. 𝑛 (𝑎𝑚)𝑛 =𝑎𝑚𝑛 𝑎 𝑆 Proof. Let 𝑥, 𝑦 ∈ A.Then, Hence, by induction on , for all in and positive integers 𝑚, 𝑛. (𝑥𝑦)2 𝑥 =𝑥2 (𝑦𝑥) = 𝑦 (𝑥2𝑥) = 𝑦 2(𝑥𝑥 )=𝑥(𝑦𝑥2). (21) 𝑛 𝑛 𝑛 Proposition 15. (𝑎𝑏) =𝑎𝑏 ,forall𝑎, 𝑏 in A and for positive 2 2 𝑛≥1 Hence, (𝑥𝑦)𝑥 = 𝑥(𝑦𝑥 ). integer .

2 2 𝑛=1 𝑛=2 Lemma 9. (𝑥 𝑦)𝑥 =𝑥 (𝑦𝑥),forall𝑥, 𝑦 in A. Proof. The result is true for .If ,then

2 2 2 2 2 Proof. Using 𝑥 𝑦=𝑦𝑥,weget (𝑎𝑏) = (𝑎𝑏)(𝑎𝑏) = (𝑎𝑎)(𝑏𝑏) =𝑎𝑏 . (26)

2 2 2 2 2 (𝑥 𝑦) 𝑥 = (𝑥𝑦) 𝑥 =𝑥 (𝑥𝑦) = (𝑦𝑥) 𝑥 =𝑥 (𝑦𝑥) . (22) Suppose that the result is true for 𝑛=𝑘. Then, we get

(𝑎𝑏)𝑘+1 = (𝑎𝑏)𝑘 (𝑎𝑏) =(𝑎𝑘𝑏𝑘) (𝑎𝑏) =(𝑎𝑘𝑎) 𝑘(𝑏 𝑏) (27) We get a more generalized form of generalized Jordan 𝑘+1 𝑘+1 algebra which is available in the following crucial theorem. =𝑎 𝑏 .

2 2 2 2 Theorem 10. (𝑥 𝑦)𝑧 =𝑥 (𝑦𝑧), 𝑥 (𝑦𝑧) =𝑥 (𝑧𝑦),forall Hence, the result is true for all positive integers. 𝑥, 𝑦, 𝑧 A in . 𝑛 𝑚 𝑚 𝑛 Theorem 16. 𝑥 𝑦 =𝑦 𝑥 ,for𝑛≥1,𝑚≥2,forall𝑥, 𝑦 in 2 2 Proof. Using 𝑥 𝑦=𝑦𝑥,weget A.

2 2 2 2 2 (𝑥 𝑦) 𝑧 = (𝑧𝑦) 𝑥 =𝑥 (𝑧𝑦) = (𝑦𝑧) 𝑥 =𝑥 (𝑦𝑧) . (23) Proof. The proof follows from Lemma 7. 𝑛 𝑚 𝑚 𝑛 Theorem 17. (𝑥 𝑥 )𝑦 = 𝑦(𝑥 𝑥 ),for𝑚 ≥ 2, 𝑛 ≥1. 6 Computational and Mathematical Methods in Medicine

Proof. Let 𝑥, 𝑦 ∈ A.Then, [4] P. Holgate, “Groupoids satisfying a simple invertive law,” The (𝑥𝑛𝑥𝑚)𝑦=(𝑥𝑚𝑥𝑛)𝑦=(𝑥𝑛𝑦) 𝑥𝑚 =𝑥𝑚 (𝑦𝑥𝑛) Mathematics Student,vol.61,no.1-4,pp.101–106,1992. [5] P. Holgate, “Sequences of powers in genetic algebras,” Journal of (28) =𝑦(𝑥𝑚𝑥𝑛). the London Mathematical Society,vol.42,pp.489–496,1967. [6] P. Holgate, “Genetic algebras associated with sex linkage,” Proceedings of the Edinburgh Mathematical Society,vol.17,pp. 113–120, 1970. Theorem 18. Every A satisfies the generalized Jordan identity 𝑚 𝑛 𝑚 𝑛 [7] P. Holgate, “Jordan algebras arising in population genetics,” (𝑥 𝑦)𝑥 =𝑥 (𝑦𝑥 ) for 𝑚≥1,𝑛≥2. Proceedings of the Edinburgh Mathematical Society. Series II,vol. 15, pp. 291–294, 1967. 𝑚=1 𝑛=2 Proof. We will use induction. For and ,itisthe [8]M.L.Reed,“Algebraicstructureofgeneticinheritance,”Ameri- same as in Theorem 17. can Mathematical Society. Bulletin. New Series,vol.34,no.2,pp. (𝑥𝑚𝑦) 𝑥𝑛 =(𝑦𝑥𝑚)𝑥𝑛 =(𝑥𝑛𝑥𝑚)𝑦=𝑦(𝑥𝑚𝑥𝑛) 107–130, 1997. [9] H. Gonshor, “Special train algebras arising in genetics,” Proceed- 𝑚 𝑛 (29) =𝑥 (𝑦𝑥 ). ings of the Edinburgh Mathematical Society,vol.12,no.1,pp.41– 53, 1960. [10] A. Worz-Busekros,¨ Algebras in Genetics,vol.36ofLecture Notes It is obvious from the above that the left almost algebra in Biomathematics, Springer, New York, NY, USA, 1980. has closed connections with generalized Jordan algebra and [11] T. Cortes, “Modular Bernstein algebras,” Journal of Algebra,vol. flexible algebra. 163,no.1,pp.191–206,1994. It is interesting to note that genetic algebras (gametic, [12] D. L. Hartl, Essential Genetics: A Genomics Perspecive,Johnand zygotic, copular, train, Bernstein, etc.) do not satisfy the prop- Bartlett Publishers, Inc., Barnstable, Mass, USA, 2002. erties of Jordan algebra but the left almost algebra becomes [13] R. D. Schafer, “Structure of genetic algebras,” American Journal a Jordan algebra. Moreover, this algebra is power associative of Mathematics,vol.71,pp.121–135,1949. 𝑚 𝑛 𝑚 𝑛 andsatisfiesthegeneralizedidentity(𝑥 𝑦)𝑥 =𝑥(𝑦𝑥 ) [14] D. Towers, “Nonassociative algebraic structures arising in 𝑛 𝑚 𝑚 𝑛 introduced by Jordan. It also satisfies 𝑥 𝑦 =𝑦 𝑥 for 𝑚, 𝑛 ≥ genetics,” The Mathematical Gazette,vol.70,no.454,pp.281– 2. This algebra is the generalization of the Jordan algebra. It 286, 1986. is noncommutative and nonassociative but possesses many characteristics similar to Jordan algebra. Since Jordan algebra has many applications in genetics, it is concluded that our new generalized algebra will give direction for applications in genetics.

3. Conclusion In this paper, we introduced a new nonassociative and non- commutative algebra with genetic realizations. We discussed its link with flexible, alternative, and Jordan algebras. This algebra possesses many characteristics similar to a commuta- tive and associative algebra. We discussed some of the genetic properties of this algebra. In our future work, we will focus on some other nonassociative algebras. We conclude that this research will give a new direction for applications in genetics.

Conflicts of Interest The authors declare that there are no conflicts of interest regarding the publication of this paper.

References

[1] I. M. H. Etherington, “Genetic algebras,” Proceedings of the Royal Society of Edinburgh,vol.59,pp.242–258,1939. [2] I. M. H. Etherington, “Non-commutative train algebras of ranks 2 and 3,” Mathematical Institute, University of Edinburgh, 241- 252. [3] I. M. Etherington, “Duplication of linear algebras,” Proceedings of the Edinburgh Mathematical Society. Series II,vol.6,pp.222– 230, 1941. M EDIATORSof INFLAMMATION

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