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 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 element. with two associative binary operations called an addition L1. x subset I of a semiring order (R, 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 ...

max{x, y}, x¯y := min{x, y}. Then we are dealing with Then (R, +, ·) is a non-commutative semiring. We can see a semiring. Indeed, suppose that x binary operation in + In this case, if x < y in R , then also x subsets of µ. 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 binary relation

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Proposition 3.8. If (R, +, ·) is a commutative semiring Proposition 3.13. Let (R, +, ·) be a semiring with x + with x + x = x for all x ∈ R, then natural number n. Let (R, +, ·) be a semiring and x, y ∈ R. Define a Proof. Assume that 2x

°c 2012 NSP Natural Sciences Publishing Cor. 102 J. S. Han, H. S. Kim and J. Neggers: Semiring Orders in a Semiring ...

Next, we perform some computations as examples in- Professor Jeong Soon Han volving the semiring order Geometry, History of Proposition 4.7. Let (R, +, ·) be a commutative semir- Mathematics, Manifold. She translated “Using History to ing and let graph theory and and applications in computer science, (World Scientific, Sin- combinatorics and include topics which are of an applied gapore, 1993). as well as a pure nature. [5] Y. B. Jun, J. Neggers and H. S. Kim, Czech. Math. J. 48 (1998), 669. [6] J. Neggers, Y. B. Jun and H. S. Kim, Czech. Math. J. 49 (1999), 127. [7] J. Neggers and H. S. Kim, Basic posets (World Scientific Pub., Singapore, 1998).

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