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Lattices in a Gamma Semiring www.ijcrt.org © 2020 IJCRT | Volume 8, Issue 11 November 2020 | ISSN: 2320-2882 LATTICES IN A GAMMA SEMIRING TILAK RAJ SHARMA Department of Mathematics, Himachal Pradesh University, Regional Centre, Khaniyara, Dharamshala, Himachal Pradesh (India)-176218 Abstract: - In this paper, the efforts are made to characterize some results on lattices of Γ −semirings. Since a complemented element plays an important role in the study of lattices, so we give characterization of some results on lattices, which are analogous to the corresponding results in semirings [2-3]. Keywords: Γ −semiring, simple, additive idempotent and multiplicative Γ −idempotent Γ − semiring, Complemented elements in a Γ − semiring, Lattices. 1. INTRODUCTION From an algebraic point of view, semirings provide the most natural common generalization of the theories of rings and most of the techniques used in analyzing semirings are taken from ring theory and group theory. The set of nonnegative integers ℕ with usual addition and multiplication provides a natural example of a semiring. There are many other examples of semiring such as the positive cone of a totally ordered ring. For a given positive integer 푛, the set of all 푛 × 푛 matrices over a semiring R forms a semiring with usual matrix addition and multiplication over R. But the situations for the set of all non-positive integers and for the set of all 푚 × 푛 matrices over a semiring R are different. They do not form semirings with the above operations, because multiplication in the above sense are no longer binary compositions. This notion provides a new kind of algebraic structure what is known as a Γ − semiring. The concept of Γ −semiring was introduced by M. M. K. Rao in 1995 [6] as a generalization of semiring as well as Γ − ring (it may be recalled here that the notion of “Γ” was first introduced in algebra by N. Nobusawa in 1964). Later it was found that Γ − semiring also provides an algebraic home to the negative cones of totally ordered rings and the sets of rectangular matrices over a semiring. For further study of semirings, Γ −semirings and their generalization, one may referred to [2-3][4-10]. IJCRT2011400 International Journal of Creative Research Thoughts (IJCRT) www.ijcrt.org 3360 www.ijcrt.org © 2020 IJCRT | Volume 8, Issue 11 November 2020 | ISSN: 2320-2882 In this paper, the efforts are made to characterize some results on lattices of Γ − semirings. Furthermore, since a complemented element plays an important role in the study of lattices, so we give characterization of some results on lattices, which are analogous to the corresponding results in semirings [2- 3]. Finally, we define harmony difference of elements of Γ푅⊥(set of all complemented elements) and proved that if d ∶ 훤푅⊥ × 훤푅⊥ → 훤푅⊥ be defined by d ∶ (x, y) → 푥∇y. Then d is metric on 훤푅⊥ with values in R. 2. PRELIMINARIES The following definitions with examples and results are felt to be an inseparable part of this paper. Definition 2.1: Let 푅 and be two additive commutative semigroups. Then 푅 is called a Γ − semiring if there exists a mapping 푅 × × 푅 → 푅 denoted by 푥훼푦 for all 푥, 푦 ∈ 푅 and 훼 ∈ satistifying the following conditions: (i) 푥훼(푦 + 푧) = (푥훼푦) + (푥훼푧). (ii) (푦 + 푧)훼푥 = (푦훼푥) + (푧훼푥). (iii) 푥 (훼 + 훽)푧 = (푥훼푧) + (푥훽푧). (iv) 푥훼(푦훽푧) = (푥훼푦)훽푧 for all 푥, 푦, 푧 ∈ 푅 and 훼, 훽 ∈ Γ. Definition 2.3: A Γ − semiring 푅 is said to have a zero element if 0훾푥 = 0 = 푥훾0 and 푥 + 0 = 푥 = 0 + 푥 for all 푥 ∈ 푅 and 훾 ∈ Γ. Definition 2.4: A non-empty subset 푇 of a Γ − semiring 푅 is said to be a sub Γ − semiring of 푅 if (푇, +) is a subsemigroup of (푅, +) and 푥훾푦 ∈ 푇 for all 푥, 푦 ∈ 푇 and 훾 ∈ Γ. Definition 2.5: A Γ − semiring is said to have identity element if 푥훾1 = 푥 = 1훾푥 for all 푥 ∈ 푅 and ∈ Γ . Definition 2.6: A Γ − semiring 푅 is said to be a commutative if 푥훾푦 = 푦훾푥 for all 푥, 푦 ∈ 푅 and 훾 ∈ Γ . Example 2.7: Let 푅 be the set of all even positive integers and Γ be set of all positive integers divisible by 3. Then with usual addition and multiplication of integers, 푅 is a commutative Γ − semiring. Definition 2.8: An element 푥 of a Γ −semiring 푅 is said to be additive idempotent if and only if 푥 + 푥 = 푥. If every element of 푅 is additive idempotent then 푅 is called additive idempotent Γ − semiring. It is denoted by 퐼+(Γ푅). Definition 2.9: An element 푥 of a Γ − semiring 푅 is said to be multiplicative Γ −idempotent if there exists 훾 ∈ Γ such that 푥 = 푥 훾푥. If every element of 푅 is multiplicative Γ − idempotent then 푅 is called multiplicative Γ − idempotent Γ − semiring. It is denoted by 퐼×(Γ푅). Definition 2.10: A Γ − semiring 푅 is said to be Γ − idempotent if it is both additive idempotent and multiplicative Γ − idempotent. We will denote the set of all Γ −idempotent elements of a Γ −semiring 푅 by 퐼(ΓR). Definition 2.11: A Γ − semiring 푅 with identity is simple if and only if 푥 + 1 = 1 = 1 + 푥 for all 푥 ∈ 푅. Definition 2.12: A Γ − semiring 푅 is centreless if and only if 푥 + 푦 = 0 implies that 푥 = 푦 = 0. IJCRT2011400 International Journal of Creative Research Thoughts (IJCRT) www.ijcrt.org 3361 www.ijcrt.org © 2020 IJCRT | Volume 8, Issue 11 November 2020 | ISSN: 2320-2882 Example 2.13: Let 푅 = ℤ+ be a set of non negative integers and Γ = {1}. Define a mapping 푅 × × 푅 → 푅 by 푥1푦 = 푥푦 for all 푥, 푦 ∈ 푅. Then 푅 is a centreless Γ −semiring.. Definition 2.14: The Centre of a Γ − semiring 푅 is a subset of 푅 consisting of all elements x of 푅 such that 푥훾푦 = 푦훾푥 for all 푦 ∈ 푅 and 훾 ∈ Γ. It is denoted by 퐶(푅). Definition 2.15: Let 푥, 푦 be elements of a Γ − semiring 푅, then 푥 is 횪 − interior 푦 denoted by 풙훁퐲 if and only if there exists an element 푧 휖 푅 such that 푥훾푧 = 푧훾푥 = 0 and 푧 + 푦 = 1 푓표푟 푎푙푙 훾 ∈ Γ. Definition 2.16: An element 푥 is complemented if and only if 푥∇x. That is, there exists an element 푦 ∈ 푅 such that 푥훾푦 = 푦훾푥 = 0 and 푥 + 푦 = 1, 푓표푟 푎푙푙 훾 ∈ Γ. This element 푦 of 푅 is the complement of 푥 in 푅. We will denote complement of 푥 by 푥⊥. Clearly if 푥 is complemented then so is 푥⊥ and 푥⊥⊥ = 푥 . Lemma 2.17[9]: Let 푅 be a Γ − semiring. Then (i) 푅 is simple if and only if 푥 = 푥 + 푥훾푦 for all 푥, 푦 ∈ 푅 and 훾 ∈ Γ. (ii) 푅 is simple if and only if 푥 = 푥 + 푦훾푥 for all 푥, 푦 ∈ 푅 and 훾 ∈ Γ. (iii) 푅 is simple if and only if 푥훾푦 = 푥훾푦 + (푥훽푧)훾푦 for all 푥, 푦, 푧 ∈ 푅 and 훽, 훾 ∈ Γ. Theorem 2.18[9]: Let 푅 be a Γ − semiring. Then every additive idempotent Γ − semiring has a simple sub Γ − semiring. Let us denote the set of all complemented elements of 푹 by 횪푹⊥. Clearly Γ푅⊥ ≠ 휙, since 0⊥=1. Indeed, if 푥 ∈ Γ푅⊥, 훾 ∈ Γ. Then 푥 + 푥⊥ = 1, 푥훾푥⊥ = 푥⊥훾푥 = 0 , for all 훾 ∈ Γ. Therefore 푥 = 푥훾1 = 푥훾(푥 + 푥⊥) = 푥훾푥 + 푥훾푥⊥ = 푥훾푥. Thus Γ푅⊥ is multiplicative Γ − idempotent, that is, Γ푅⊥ ⊆ 퐼×(Γ푅). Again let 푥 ∈ Γ푅⊥ 푎푛푑 훾 ∈ Γ. Set 푥 ⊛ 푦 = 푥 + 푥⊥훾푦, for all ∈ Γ . Then 푥 ⊛ 푥⊥ = 푥 + 푥⊥훾푥⊥ = 푥 + 푥⊥ = 1 for all 푥 ∈ Γ푅⊥. Also, if 푥 + 푦 = 1 then 푥⊥ = 푥⊥훾1 = 푥⊥훾(푥 + 푦) = 푥⊥훾푥 + 푥⊥훾푦 = 푥⊥훾푦. Thus 푥 ⊛ 푦 = 푥 + 푥⊥훾푦 = 푥 + 푥⊥ = 1. Lemma 2.19[10]: Let 푅 be a Γ − semiring such that 푥, 푦 ∈ Γ푅⊥. Then x훾푦, 푥 ⊛ 푦 ∈ Γ푅⊥. Lemma 2.20[10]: Let 푅 be a Γ − semiring and 푥, 푦 ∈ Γ푅⊥. Then 푥훾푦 = 푦훾푥, 푓표푟 푎푙푙 훾 ∈ Γ. Lemma 2.21[10]: Let 푅 be a centreless Γ − semiring. If 푥, 푦 ∈ Γ푅⊥ and β, 훾 ∈ Γ. Then (푥훾푦)훽푥⊥ = (푥⊥훾푦)훽푥 = 0. Theorem 2.22[10]: Let 푅 be a centreless Γ − semiring. Then Γ푅⊥ is Γ − idempotent, commutative, simple Γ − semiring with the mapping ⊛ : Γ푅⊥ × Γ × Γ푅⊥ → Γ푅⊥ defined by 푥 ∗ 훾 ∗ 푦 = 푥훾푦, for all 푥, 푦 ∈ Γ푅⊥, 훾 ∈ Γ. Proposition 2.23[10] Let 푅 be a centreless Γ −semiring. Then Γ푅⊥is a sub Γ −semiring of 푅 if and only if 푥 + 푦, 푥훾푦 ∈ Γ푅⊥, 푥, 푦 ∈ Γ푅⊥. Remark 2.24: Throughout this paper, 푅 will denote a Γ − semiring with zero element ′ퟎ′ and identity element ′ퟏ′ unless otherwise stated. IJCRT2011400 International Journal of Creative Research Thoughts (IJCRT) www.ijcrt.org 3362 www.ijcrt.org © 2020 IJCRT | Volume 8, Issue 11 November 2020 | ISSN: 2320-2882 3. LATTICES IN A GAMMA SEMIRING Complemented elements play an important role in study of lattices. Another major source of inspiration for the theory of Γ − semirings is lattice theory. Definition 3.1: Let 푎 and 푏 be two elements in a partially ordered set (퐴, ≤). An element 푐 is said to be an upper bound of 푎 and 푏 if 푎 ≤ 푐 and 푏 ≤ 푐, and an element 푐 is said to be a least upper bound of 푎 and 푏 if 푐 is an upper bound of 푎 and 푏, and there is no other upper bound 푑 of 푎 and 푏 such that 푑 ≤ 푐.
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