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Semiring Orders in a Semiring -.:: Natural Sciences Publishing Appl. Math. Inf. Sci. 6, No. 1, 99-102 (2012) 99 Applied Mathematics & Information Sciences An International Journal °c 2012 NSP Natural Sciences Publishing Cor. Semiring Orders in a Semiring Jeong Soon Han1, Hee Sik Kim2 and J. Neggers3 1 Department of Applied Mathematics, Hanyang University, Ahnsan, 426-791, Korea 2 Department of Mathematics, Research Institute for Natural Research, Hanyang University, Seoal, Korea 3 Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350, U.S.A Received: Received May 03, 2011; Accepted August 23, 2011 Published online: 1 January 2012 Abstract: Given a semiring it is possible to associate a variety of partial orders with it in quite natural ways, connected with both its additive and its multiplicative structures. These partial orders are related among themselves in an interesting manner is no surprise therefore. Given particular types of semirings, e.g., commutative semirings, these relationships become even more strict. Finally, in terms of the arithmetic of semirings in general or of some special type the fact that certain pairs of elements are comparable in one of these orders may have computable and interesting consequences also. It is the purpose of this paper to consider all these aspects in some detail and to obtain several results as a consequence. Keywords: semiring, semiring order, partial order, commutative. The notion of a semiring was first introduced by H. S. they are equivalent (see [7]). J. Neggers et al. ([5, 6]) dis- Vandiver in 1934, but implicitly semirings had appeared cussed the notion of semiring order in semirings, and ob- earlier in studies on the theory of ideals of rings ([2]). tained some results related with the notion of fuzzy left Semirings occur in different mathematical fields, i.e., as ideals of the semirings. ideals of a ring, as positive cones of partially ordered rings Suppose that R is a semiring. Define a relation <R on and fields, in the context of topological considerations, and R as follows: in the foundations of arithmetic, including questions raised by school education. Semirings have become of great in- x < y provided x + y = y and xy = x; x 6= y: terest as a tool in different branches of computer science R ([4]). Thus, since 0 + y = y and 0y = 0, it follows that if y 6= 0, By a semiring ([1]) we shall mean a set R endowed then 0 <R y always, i.e., 0 is a unique miminal element. with two associative binary operations called an addition L1. x <R y and y <R x is impossible. and a multiplication (denoted by + and ¢, respectively) sat- L2. x <R y and y <R z implies x <R z. isfying the following conditions: The set (R; <R) is a poset with unique miminal ele- (i).addition is a commutative operation, ment 0. We shall refer to it as the semiring order of R.A (ii).there exists 0 2 R such that x + 0 = x and x0 = 0x = non-empty subset I of a semiring order (R; <R) is called 0 for each x 2 R, and an order ideal if x 2 I; y <R x imply y 2 I. + (iii).multiplication distributes over addition both from the Example 2.1. Let R be the collection of non-negative left and from the right. real numbers with the usual operations \+" and \¢". Then + (R ; +; ¢) is a semiring. Also, if x <R+ y then x + y = y means x = 0 and y 6= 0. In particular, if x 6= y, and 1. Preliminaries x 6= 0; y 6= 0, then x ± y, i.e., x and y are incomparable. + Hence, (R ¡f0g; <R+ ) is an antichain. We shall consider There are two ways to define a partially ordered set on a R+ to be an antichain semiring. set: (i) weak inclusion: reflexive, anti-symmetric and tran- Example 2.2. Let R+ be the collection of non-negative sitive; (ii) strong inclusion: irreflexive and transitive, and real numbers. Define operations \©" and \¯" by x©y := ¤ Corresponding author: e-mail: [email protected] °c 2012 NSP Natural Sciences Publishing Cor. 100 J. S. Han, H. S. Kim and J. Neggers: Semiring Orders in a Semiring ... maxfx; yg, x¯y := minfx; yg. Then we are dealing with Then (R; +; ¢) is a non-commutative semiring. We can see a semiring. Indeed, suppose that x <R+ y. Then x © y = that 2 <¢ 3; 3 <¢ 2, but not 3 <¢ 3, i.e., <¢ is not a partial + y and x ¯ y = x. If r 2 R , then x <R+ y implies order on R. r ¯ x <R+ r ¯ y as well. Hence (r ¯ x) © (r ¯ y) = Even though we obtained the posets as in Propositions r ¯ y = r ¯ (x © y). Thus (R+; ©; ¯) is a semiring. 3.1 and 3.2, they were made by just one binary operation in + In this case, if x < y in R , then also x <R y in the semirings, while semirings were defined by two binary op- semiring order, whence the two orders are the same since erations. This means the partial orders discussed in Propo- an order extension of a chain is precisely the chain itself. sition 3.1 and 3.2 have some defects. + Thus, we shall consider (R ; ©; ¯) to be a chain semiring. The semiring order <R discussed in section 2 is an in- J. Neggers et. al. [6] obtained that if ¹ : R ! L is an tersection of <+ and <¢ in a (not necessarily commutative) L-fuzzy left ideal of the semiring (R; +; ¢), then ¹t is an semiring, i.e., <R=<+ \ <¢. order ideal of (R; <R) and any finite order ideal I is a level Proposition 3.4. Let (R; +; ¢) be a semiring with xy = subset of ¹. Moreover, they proved that if ¹ : R ! L is an x ) x + y = y; 8x; y 2 R. Then <¢ is a partial order on L-fuzzy left ideal of a finite chain semiring (R; +; ¢) then R. the collection of order ideals of (R; <R) is the collection Proof. Let x <¢ y; y <¢ z. Then xy = x; yz = y; x 6= of level subsets of ¹. Furthermore, this collection is lin- y; y 6= z. By assumption x+y = y; y +z = z. This means early ordered by set inclusion. This paper is a continuation that x <R y; y <R z. Since <R is a partial order, x <R z, of [6] on the study of semiring orders in a semiring. proving x 6= z. Hence x <¢ z. We define another semiring order on commutative semirings as follows. Theorem 3.5. Let (R; +; ¢) be a commutative semiring 2. Some semiring orders and x; y 2 R. Define a binary relation <B on R by x <B y if and only if x + y + xy = y; xy = x; x 6= y. Then (R; < ) In this section we introduce several semiring orders in B is a poset. x < y y < z x + y + xy = semirings, and investigate some relations between them. Proof. Let B and B . Then y; xy = x; x 6= y and y + z + yz = z; yz = y; y 6= z, Proposition 3.1. Let (R; +; ¢) be a semiring and x; y 2 and hence xz = (xy)z = x(yz) = xy = x. We claim R. If we define a relation < on R by x < y if and only + + that x 6= z. Assume that x = z. Then z = y + z + yz = if x + y = y; x 6= y, then it is a partial order. y + x + yx = x + y + xy = y, a contradiction. Moreover, Proof. Clearly, <+ is irreflexive. If x <+ y; y <+ z, then x + y = y; y + z = z; x 6= y, and y 6= z. Hence x + z + xz = x + (y + z + yz) + xz x + z = x + (y + z) = (x + y) + z = y + z = z. We = (x + y + xz) + z + yz claim that x 6= z. Assume that x = z. Then z = y + z = = (x + y + x) + z + yz y + x = x + y = y, a contradiction. Hence x <+ z, proving the proposition. ut = (x + y + xy) + z + yz Since 0 + x = x for any x 2 R, 0 <+ x for any x 2 = y + z + yz R ¡ f0g. Hence (R; <+) is a poset with unique minimal = z: element 0. Hence x < z. Proposition 3.2. Let (R; +; ¢) be a commutative semir- B Proposition 3.6. If (R; +; ¢) is a commutative semir- ing and x; y 2 R. If we define a relation < on R by x < y ¢ ¢ ing, then < ⊆< . if and only if xy = y; x 6= y, then it is a partial order. R B Proof. If (x; y) 2< , then x + y = y; xy = x; x 6= y. < x < y; y < z R Proof. Clearly, ¢ is irreflexive. If ¢ ¢ , then Hence x + y + xy = x + y + x = y + x = x + y = y, xy = x; yz = y xz = (xy)z = x(yz) = , and hence proving (x; y) 2< . xy = x x 6= z x = z B . We claim that . Assume that . Then Note that < = < does not hold in non-commutative x = xy = zy = yz = y (R; ¢) R B , a contradiction, since is rings in general. See the following example. commutative. Example 3.7. Let R := f0; 1; 2; 3g be a set with the Since 0x = 0 for any x 2 R, 0 <¢ x for any x 2 following tables: R ¡ f0g. Hence (R; <¢) is a poset with unique minimal element 0. Example 3.3. Let R := f0; 1; 2; 3g be a set with the + 0 1 2 3 ¢ 0 1 2 3 following tables: 0 0 1 2 3 0 0 0 0 0 1 1 1 2 3 1 0 0 1 1 2 2 2 2 3 2 0 1 2 2 + 0 1 2 3 ¢ 0 1 2 3 3 3 3 3 2 3 0 1 3 3 0 0 1 2 3 0 0 0 0 0 1 1 1 2 3 1 0 1 1 1 Then (R; +; ¢) is a non-commutative semiring.
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