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Semi Unit Graphs of Commutative Semi Rings Communications in Mathematics and Applications Vol. 10, No. 3, pp. 519–530, 2019 ISSN 0975-8607 (online); 0976-5905 (print) Published by RGN Publications http://www.rgnpublications.com DOI: 10.26713/cma.v10i3.1203 Research Article Semi Unit Graphs of Commutative Semi Rings Yaqoub Ahmed* and M. Aslam Department of Mathematics, GC University, Lahore, Pakistan *Corresponding author: [email protected] Abstract. In this article, we introduce semi unit graph of semiring S denoted by »(S). The set of all elements of S are vertices of this graph where distinct vertices x and y are adjacent if and only if x y is a semiunit of S. We investigate some of the properties and characterization results on Å connectedness, distance, diameter, girth, completeness and connectivity of »(S). Keywords. Semirings; Semiunits; k-ideals; Graphs MSC. 16Y60; 05C25 Received: February 26, 2019 Revised: May 1, 2019 Accepted: Accepted 13, 2019 Copyright © 2019 Yaqoub Ahmed and M. Aslam. 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. 1. Introduction The concepts of the graph to the elements (zero-divisors) of a ring was introduced by Beck in [8] and discussed the coloring of a commutative ring. The zero-divisor graph of a commutative ring has been studied extensively by several authors like Anderson [2], and Atani [3,5]. Moreover, Atani and others establish the properties of zero divisor graphs in semirings and introduced unit graph in semirings in [3,5]. In this paper, our focus is on semi unit elements of semirings and their graphs. For the sake of completeness, we state some definitions and notations used throughout this paper. In this paper, we refer S to be commutative semi ring with multiplicative identity and absorbing zero, unless mentioned otherwise. A non-zero element a S is said to be semi unit 2 if there exists, r, s S such that 1 r.a s.a (c.f. [7]). The set of all semi units of S is denoted 2 Å Æ by Su and the set of all non-semi units is denoted by Nu in this paper. Every unit is a semi 520 Semi Unit Graphs of Commutative Semi Rings: Y. Ahmed and M. Aslam unit, by taking r 0 in definition. In a ring every semi unit is a unit. An ideal K is said to Æ be k-ideal (Subtractive ideal) such that if x, x y K then y K (c.f. [10]). A k-ideal which is Å 2 2 also maximal ideal is called maximal k-ideal. P is maximal k-ideal of S if and only if S/P is a semi field [7]. Let P be an ideal of a semi ring S, P is a prime k-ideal of S if and only if S/P is a semi domain [4]. Let S be a semi ring with non-zero identity. S is said to be a local semi ring if and only if S has a unique maximal k-ideal. The k-closure cl(I) of ideal I is defined by cl(I) {a S;a c d for some c, d I} which is indeed the smallest k-ideal of S containing Æ 2 Å Æ 2 I [10]. An ideal I of semiring S is called a partitioning ideal(Q-ideal) if there exists a subset Q of S such that: S {q I : q Q}. If q1, q2 Q then (q1 I) (q2 I) if and only if q1 q2. If I Æ[ Å 2 2 Å \ Å 6Æ Æ is a Q-ideal of a semiring S then I is a k-ideal of S, by [12, Lemma 2]. For basic concepts of semirings, we refer [1, 9, 11]. There are some special families of graphs as complete graph is a simple graph in which any two vertices are adjacent. A graph is connected if there is a path between any two vertices. A graph is bipartite if its vertex set can be partitioned into two subsets X and Y so that every edge has one end in X and one end in Y . If » is simple and every vertex in X is joined to every vertex in Y , then » is called a complete bipartite graph. An isomorphism between two simple graphs X and Y as a bijection θ : V (X) V (Y ) which preserves adjacency. For undefined terms ! related to graph theory, we refer [16]. 2. Results of Semi Units Firstly, we discuss some results on semiunit elements, which are helpful to study the graphs of semiunits. Lemma 2.1 ([7]). Let S be a semiring and let a S. Then a is a semi-unit of S if and only if a 2 lies outside each k-maximal ideal of S. Lemma 2.2 ([7]). If x S be a commutative semi ring, then cl(Sx) is a k-ideal of S. 2 Proposition 2.3. Every ideal generated by a non-semi unit in a commutative semiring S is proper ideal and contain in some k-maximal ideal. Proof. Consider a be a non-semi unit in S. It is observed that Sa is an ideal. Also, Sa cl(Sa) ⊆ which is k-ideal, whereas every k-ideal of semi ring contained in some k-maximal ideal by ([15, Corollary 2.2]), so there must exist some k-maximal ideal M (say) such that a Sa h i Æ ⊆ cl(Sa) M implies that a Sa M. ⊆ h i Æ ⊆ Proposition 2.4. The set of semi units Su of semi ring S are closed under multiplication. Proof. Consider a, b Su, such that 2 1 r.a s.a and 1 t.b u.b for some r, s, t, u S . Å Æ Å Æ 2 Communications in Mathematics and Applications, Vol. 10, No. 3, pp. 519–530, 2019 Semi Unit Graphs of Commutative Semi Rings: Y. Ahmed and M. Aslam 521 Let a.b is a non-semiunt. By using Proposition 2.1, we get that there exist a proper ideal ab , h i which contain in some k-maximal ideal M such that ab M implies that ab M. M is an h i ⊆ 2 ideal therefore sab, rab M, for all r, s S. Also, we have 1 r.a s.a or b r.a.b s.a.b M 2 2 Å Æ Å Æ 2 this implies that b r.a.b M, since M is k-ideal therefore b M which is a contradiction. Å 2 2 Theorem 2.5. Let the set of all non-semiunits Nu is not an ideal then Nu is not closed under addition. Proof. By hypothesis, we have Nu is not an ideal. Let a S and x Nu. We will show that a.x Nu. Suppose on contrary that a.x Su. It gives 2 2 2 2 1 (a.x).r (a.x).s for some r, s S Å Æ 2 that is 1 x.(a.r) x.(a.s) or 1 x.r1 x.s1, where r1 a.r, s1 a.s S, this implies that x Å Æ Å Æ Æ Æ 2 is semi unit, which is contradiction. Therefore, a.x Nu. Then by hypothesis, we have some 2 distinct non-semi units x, y Nu such that x y Su. 2 Å 2 3. Semi Unit Graphs Consider S is commutative semi ring and Su denotes of set of all semi unit elements in S. We develop a semi unit graph, denoted by », by taking every element as vertex. If x, y S (x y) 2 6Æ then x and y are adjacent if x y Su, otherwise they are disjoint vertices therefore we are Å 2 considering simple graphs. We shall discuss about its properties, characteristics and different shapes of Semi Unit graphs in this section. Theorem 3.1. Consider a Semi ring S, then the Semi unit graph is complete if and only if Su S {0}. Æ ¡ Proof. If Su S {0}, then Su is zero-sum free, since S is semi ring otherwise it will be a ring Æ ¡ (c.f. [6, Lemma 2.1]). This shows that Su is closed under addition, so every two vertices of Su are adjacent. Also 0 is adjacent with all other elements of S as 0 x x Su, therefore graph is Å Æ 2 complete. Conversely, suppose » is complete graph then every vertex x S of graph is connected with 2 all other vertices. As 0 S, therefore 0 is also adjacent to every x S and 0 x x then by 2 2 Å Æ definition of semiunit graph x Su, this implies that all non-zero elements x Su, that is 2 2 Su S {0}. Æ ¡ Theorem 3.2. Let S be a Semi ring with multiplicative identity 1. If Nu makes k-ideal then » is connected graph. Proof. Consider S is semi ring with and the set of all non-semi units Nu makes k-ideal. Let x, y Nu, then x y Nu, that there is no edge between them. Now consider x Nu, and u Su 2 Å 2 2 2 then x u must be semi unit. Suppose on contrary that x Nu, and x u Nu then by hypothesis Å 2 Å 2 u Nu which is contradiction, so x u Su, there is always an edge between a semi unit and 2 Å 2 non- semi unit in this semi ring, so all vertices must be connected at least through identity 1.
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