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Quotient Seminear-Rings ISSN (Print) : 0974-6846 Indian Journal of Science and Technology, Vol 9(38), DOI: 10.17485/ijst/2016/v9i38/89115, October 2016 ISSN (Online) : 0974-5645 Quotient Seminear-Rings Fawad Hussain1, Muhammad Tahir2, Saleem Abdullah2 and Nazia Sadiq2 1Department of Mathematics, Abbottabad University of Science and Technology, Pakistan; [email protected] 2Department of Mathematics, Hazara University Mansehra, Pakistan; [email protected]; [email protected]; [email protected] Abstract The study of a seminear-ring was started in 1967. Seminear-ring is the generalization of semiring and nearing. It is known that in a semiring the quotient structure is constructed by congruence relations through c-ideals and homomorphisms. We apply the said congruence relations to a seminear-ring and get quotient structure of a seminear-ring. The aim of this paper quotientis to discuss seminear-rings. quotient seminear-rings in two different ways. We study congruence relations, homomorphisms and ideals. We show that each homomorphism and c-ideal define a congruence relation on seminear-rings. At the end, we discuss Keywords: Seminear-ring, c-Ideals, Homomorphism, Congruence Relation, Quotient Seminear-Ring. 1. Introduction rings and explored some interesting properties. There is another intersecting paper7 in which the author discussed The study of semirings was started by the German seminear-rings as well as semi-near fields. Further differ- Mathematician Dedekind1. They were later studied by ent people in8-11 worked on seminear-rings and explored different Mathematicians, particularly by the American many interesting and elegant properties. The author of8,9 Mathematician Vandiver. He did work very hard on discussed substructures in seminear-rings. The author of11 semirings. He wanted to accept a semiring as fundamen- discussed radicals in seminear-rings while the authors’ tal and best algebraic structure2. He was not successful of10 discussed weekly regular seminear-rings. Before because of few reasons and semirings had fallen into four years ago Perumal and Balakrishnan discussed disuse. However, during the late 1960’s real and sig- left bipotent seminear-rings in12 and discussed left duo nificant applications were found of semirings in several seminear-rings in13 and explored some useful properties. fields. These fields include: automata theory, optimiza- In this paper, we study quotient structure of a seminear- tion theory, graph theory, the theory of discrete event ring while different quotient algebraic structures have dynamical system, coding theory, analysis of computer been studied in14,15. programs, algebras of formal processes and generalized fuzzy computation. The detail of these can be found 2. Preliminaries in3. Later on, different people worked on semirings and explored many interesting properties of semirings. In4 the This portion contains some of the basic definitions, fun- rough set theory has been applied to semirings while in5 damental results, and reviews some of the background some applications of a semiring have been studied. In the material, which will be used in the coming sections. In paper6 the authors introduced the notion of a seminear- 1967, Hoorn and Rootoselaar introduced the notion of a ring which was a generalization of a semiring and then semiring which was a generalization of a semiring. We explored some properties of seminear-rings. Especially start with the following definition of a semiring which has the authors of6 discussed homomorphisms in seminear- been taken from7. *Author for correspondence Quotient Seminear-Rings 2.1 Definition (i) x + y S for all x, y S, (ii) x ⋅ y S for all x, y S. A seminear-ring is a non-empty set R together with two ∈ ∈ binary operations ‘+’ and ‘ ’ such that (R, +) and (R, ) are Proof ∈ ∈ semigroups such that The proof is easy and is left for the readers. (a + b) ⋅ c = (a ⋅ c) + (b ⋅ c) holds for all a, b, c R. We are now going to define ideals. The following defi- nition has been taken from the paper8. It will be more suitable to call what we have just∈ defined a right seminear-ring. In the same way, we may define a 2.5 Definition left seminear-ring in which one use left distributive law instead of right distributive law. In this paper, we shall A non-empty subset I of a seminear-ring(R, +, ) is called keep to right seminear-rings throughout. To understand a left (respectively right) ideal ofR if the following condi- the notion of semirings, we give some examples. tions hold: (i) x + y ∈ I for all x, y ∈ I, 2.2 Examples (ii) a ⋅ x ∈ I (x ⋅ a ∈ I ) for all x ∈ I and a ∈ R. I is called a two-sided ideal or simply an ideal of R if it (i) All rings and semirings are seminear-rings. is both left as well as right ideal of R. (ii) Let N be the set of natural numbers, then N is a Note that it is obvious from the above definition that semiring with ordinary addition and multiplication every ideal is a subseminear-ring and we know that every of numbers. subseminear-ring is a seminear-ring. Thus, it follows that We are now giving a non-trivial example of a semi- every ideal is a seminear-ring. near-ring. We are now going to define c-ideals in seminear- (iii) Let N be the set of natural numbers. Let rings. The idea of this definition comes from the paper16 a b in which the authors define c-ideals in semirings. Ω = : a, b, c, d ∈ N c d 2.6 Definition and define, A+ B = AB and A ⋅ B = A for all A, B Ω . An ideal I of a seminear-ring(R, +, ⋅) is called a c-ideal Then ( Ω , +, ⋅) is a seminear-ring. of R if for any a, b ∈ R, there exist x, y ∈ I such that We are now going to define subseminear-rings and a + b + x = y + b + a. ideals in seminear-rings. Firstly we are going to define Like homomorphisms of other algebraic structures subseminear-rings for which we don’t have a specific such as semirings and rings, homomorphisms of semi- reference. near-rings are those maps which preserve the binary operations. We now give a proper definition of a semi- 2.3 Definition near-ring homomorphism. The following definition is taken from8. A non-empty subset S of a seminear-ring (R, +, ⋅) is called a subseminear-ring of R if S itself is a seminear-ring under the operations of R when restricted to S. 2.7 Definition It should be noted that very seminear-ring (R, +,⋅) has Let (R, +, ∙) and (S, , ) be two seminear-rings. A at least one subseminear-ring, i.e. R itself called the trivial mapping µ : R S from R to S is called a homomor- subseminear-ring. All other subseminear-rings are called phism if and only if: non-trivial subseminear-rings. We are now going to pres- (i) µ (a + b) = µ (a) µ(b) ent some properties of subseminear-rings. The result (ii) µ (a ⋅ b) = µ (a) µ(b) given below gives us necessary and sufficient conditions for all a, b R. for subseminear-rings and for which we don’t have a spe- It should be noted that if R = S, then the cific reference. homomorphism∈ µ is known as an endomorphism. If µ is homomorphism as well as onto, then µ is called an 2.4 Theorem epimorphism. If µ is homomorphism as well as one-one, Let (R, +, ⋅) be a seminear-ring, then a non-empty subset then µ is called a monomorphism. A homomorphism µ S of R is called a subseminear-ring of R if and only if is called an isomorphism if it is epimorphism as well as 2 Vol 9 (38) | October 2016 | www.indjst.org Indian Journal of Science and Technology Fawad Hussain, Muhammad Tahir, SaleemAbdullah and Nazia Sadiq ρ monomorphism. An isomorphism µ is called an auto- Similarly one can show that is a right congruence morphism if R = S. If µ : R S is an isomorphism then relation. ρ we say that R is isomorphic to S and write: Conversely, suppose that is both left as well as right congruence relation. Choose s, t, u, v ∈R such that (s, t), R S. (u, v)∈ρ. This implies that (s + u, t + u), (s ∙ u, t ∙ u)∈ρ, because ρ is right compatible and (t + u, t + v), (t ∙ u, t ∙v)∈ρ, 3. Congruence Relations because ρ is left compatible. It follows that s( + u, t + v), (s ∙ u, t ∙ v)∈ρ, because ρ is transitive. This completes the This section concerned with the congruence relations on proof. seminear-rings. We define congruence relations and show that each homomorphism defines a congruence relation 4. Congruence Relations and on seminear-rings. The idea of this section comes from the book17 in which the author does similar calculations Homomorphisms for semigroups. In this section we state and prove a result which says that corresponding to every homomorphism there is a con- 3.1 Definition gruence relation. The result is important because once we Let (R, +, ∙) be a seminear-ring. A relation ρ on R is called get it we can get quotient seminear-rings. At the end we left compatible if for all s, t, a∈R such that (s, t) ∈ρ prove analogues of the isomorphism theorems. implies that (a + s, a + t), (a ∙ s, a ∙ t)∈ρ. ρ is called right compatible if for all s, t, b ∈R such 4.1 Theorem that (s, t) ∈ρ implies that (s + b, t + b), (s ∙ b, t ∙ b) ∈ ρ. If h is a homomorphism from a seminear-ring (R, +, ∙) ρ is called compatible if for all s, t, u, v∈R such that (s, to a seminear-ring (S, , ), then h defines a congru- t), (u, v) ∈ρimplies that (s + u, t + v), (s ∙u, t∙v)∈ρ.
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