International Journal of Pure and Applied Volume 120 No. 7 2018, 185-193 ISSN: 1314-3395 (on-line version) url: http://www.acadpubl.eu/hub/ Special Issue http://www.acadpubl.eu/hub/

A Study of Sub-Distributive

Rosy Joseph Associate Professor in Mathematics & Controller of Examinations, Stella Maris College (Autonomous), Chennai 600086, Tamil Nadu, India

Abstract From an algebraic point of view, Semirings provide the most natural generalization of theory and theory. In the absence of additive inverse in a , one had to impose a weaker condition on the semiring, i.e., the additive cancellative law to study interesting structure properties. In many practical situations, fuzzy numbers are used to model imprecise observations derived from uncertain mea- surements or linguistic assessments. In this connection, a special class of fuzzy numbers whose shape is symmetric with respect to a vertical line called the symmetric fuzzy numbers i.e., for α (0, 1] the α cuts will have a constant ∈ − mid-point and the upper end of the interval will be a non- increasing function of α, the lower end will be the image of this function, is suitable. It was observed that the structure of the class of symmetric fuzzy numbers forms a commuta- tive semigroup with cancellative property. Also it forms a multiplicative monoid satisfying sub-distributive property. In this paper we introduce the , sub- distributive semiring and discuss its various properties viz., ideals and prime ideals of sub-distributive semiring, sub- distributive ring of difference etc. in detail.

Keyword: Semirings, Symmetric fuzzy numbers, sub-distributive semiring, sub-distributive ring of difference.

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1 Introduction

Semirings were first considered explicitly by H. S. Vandiver in 1935 and has since then been studied by many authors. Semirings consti- tute a fairly natural generalization of rings, with broad applications in the mathematical foundations of computer science. J.S. Golan introduced the concept of ring of differences R∆ where R is an additively cancellative semiring. Ram Parkash Sharma and Rosy Joseph [3] have obtained various properties of ideals of a semiring R and its ring of differences R∆. In this paper the author introduces an algebraic structure, namely, sub-distributive semiring which is obtained by dropping the distributive property from semirings.

2 Preliminaries

For the sake of completeness, we state some definitions from [1] which are necessary for the development of this paper.

Definition 1. A semiring is a nonempty R on which oper- ations of and have been defined such that the following conditions are satisfied:

1.( R, +) is a commutative monoid with 0;

2.( R,.) is a monoid with identity 1;

3. Multiplication distributes over addition from either side;

4.0 r = 0 = r0 for all r R; ∈ 5.1 = 0. 6 Remark 2. A semiring (R, +,.) is partially-ordered if and only if there exists a partial order relation on R satisfying the ≤ following conditions for elements r, r0, r00 of R

1. If r r0 then r + r00 r0 + r00 ≤ ≤

2. If r r0 and r00 0 then rr00 r0r00 and r00r r00r0 ≤ ≥ ≤ ≤ If the relation is a , then R is totally-ordered. ≤

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Definition 3. A semiring R is said to be yoked if for a, b R there exists an element r R such that a + r = b or b + r = a.∈ ∈ Definition 4. A left ideal I of a semiring R is a nonempty subset of R satisfying the following conditions:

1. If a, b I, then a + b I; ∈ ∈ 2. If a I and r R, then ra I; ∈ ∈ ∈ 3. I = R. 6 Remark 5. A right ideal of R is defined in the analogous manner and an ideal of R is a subset which is both a left and a right ideal of R.

Definition 6. An ideal I of a semiring R is prime if AB I implies A I or B I for any ideals A, B of R. ⊆ ⊆ ⊆ Definition 7. A semiring R is called additively cancellative if for each a R, a + b = a + c implies that b = c in R. ∈ Remark 8. If a semiring R is additively cancellative, then the ring of differences R∆ exists. We write R∆ = a b a, b R for the ring of difference. For a b, c d { R−∆, we| have∈ } − − ∈ (a b) + (c d) = (a + c) (b + d) − − − (a b)(c d) = (ac + bd) (ad + bc) − − − For more details see [1]. For any ideal A of a semiring R, A∆ is defined as A∆ = a b a, b A . Clearly A∆ is an ideal of R∆. { − | ∈ } 3 Sub-Distributive Semiring

This section deals with an algebraic structure similar to semirings except that it does not hold the distributive property. We start with

Definition 9. A sub-distributive semiring is a nonempty set (R, +,.) satisfying the following conditions:

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1.( R, +) is a commutative monoid with identity 0;

2.( R,.) is a monoid with identity 1;

3. a (b + c) ab + ac for all a, b, c R; ≤ ∈ 4.0 r = 0 = r0 for all r R; ∈ 5.1 = 0. 6 Definition 10. A sub-distributive semiring R is said to be commutative if it is commutative under multiplication.

Before proceeding further, we give an example of a sub-distributive semiring.

Example 11. Let R be the set of symmetric fuzzy num- bers and A,˜ B˜ be two symmetric fuzzy numbers with α cuts ˜ α α ˜ α α − [A]α = [a1 , a2 ] , [B]α = [b1 , b2 ]. Following [4], addition and multiplication are defined as follows: Addition aα + aα bα + bα aα aα bα bα [A˜ + B˜] = 1 2 + 1 2 + 2 − 1 + 2 − 1 [ 1, 1] α 2 2 2 2 −   Multiplication aα + aα bα + bα [A.˜ B˜] = 1 2 . 1 2 + α 2 2 aα aα bα + bα aα + aα bα bα aα aα bα bα 2 − 1 1 2 + 1 2 2 − 1 + 2 − 1 2 − 1 [ 1, 1] 2 2 2 2 2 2 −  

Ordering aα + aα bα + bα A˜ B˜ if either 1 2 < 1 2 ≺ 2 2 aα + aα bα + bα aα aα bα bα or 1 2 = 1 2 and 2 − 1 < 2 − 1 2 2 2 2 aα + aα bα + bα aα aα bα bα A˜ = B˜ if 1 2 = 1 2 and 2 − 1 = 2 − 1 2 2 2 2

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Then (R, +,.) becomes a partially-ordered, commutative monoid with 0˜ and multiplicative monoid with identity 1.˜ aα + aα aα aα bα + bα cα + cα [A.˜ (B˜ + C˜)] = 1 2 + 2 − 1 [ 1, 1] . 1 2 + 1 2 α 2 2 − 2 2    bα bα cα cα + 2 − 1 + 2 − 1 [ 1, 1] 2 2 −    aα + aα bα + bα cα + cα = 1 2 . 1 2 + 1 2 2 2 2   aα aα bα + bα cα + cα + 2 − 1 1 2 + 1 2 2 2 2  a α + aα bα bα cα cα + 1 2 2 − 1 + 2 − 1 2 2 2   aα aα bα bα cα cα + 2 − 1 2 − 1 + 2 − 1 [ 1, 1] 2 2 2 −   aα + aα bα + bα aα + aα cα + cα 1 2 1 2 + 1 2 1 2 +  2 2 2 2 aα aα bα + bα aα aα cα + cα 2 − 1 1 2 + 2 − 1 1 2 2 2 2 2  aα + aα bα bα aα + aα cα cα + 1 2 2 − 1 + 1 2 2 − 1 2 2 2 2

aα a α bα bα aα aα cα cα + 2 − 1 2 − 1 + 2 − 1 2 − 1 [ 1, 1] 2 2 2 2 −  aα + aα bα + bα aα aα bα + bα aα + aα bα bα = 1 2 1 2 + 2 − 1 1 2 + 1 2 2 − 1 2 2 2 2 2 2  aα aα bα bα + 2 − 1 2 − 1 [ 1, 1] 2 2 −  aα + aα cα + cα aα aα cα + cα aα + aα cα cα + 1 2 1 2 + 2 − 1 1 2 + 1 2 2 − 1 2 2 2 2 2 2  aα aα cα cα + 2 − 1 2 − 1 [ 1, 1] 2 2 −  = [A.˜ B˜]α + [A.˜ C˜]α

Thus the set of symmetric fuzzy numbers form a sub-distributive semiring, which is clearly commutative and cancellative in both addition and multiplication.

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Remark 12. If a sub-distributive semiring is additively can- cellative, then we can define the sub-distributive ring of differences analogous to ring of differences.

Lemma 13. Let R be a sub-distributive semiring and R∆ its sub-distributive ring of differences. Let A, B be two ideals of R and P,Q two ideals of R∆. Then 1. (a) A AM R. Equality holds if A is subtractive; ⊆ ∩ (b) A B implies AM BM; ⊆ ⊆ 2.( P R)(Q R) (PQ) R; ∩ ∩ ⊆ ∩ 3. If A and B are subtractive, then (a) AM BM implies A B ; ⊆ ⊆ (b) AM ( BM implies A ( B; 4.( P R)M P. Equality holds if R is yoked; ∩ ⊆ 5. If P Q, then P R Q R ⊆ ∩ ⊆ ∩ Proof. Given R a sub-distributive semiring and R∆ its sub- distributive ring of differences, A, B two ideals of R and P,Q two ideals of R∆ 1. (a) Since AM = a b a, b A , clearly A AM R. As- { − | ∈ } ⊆ ∩ sume A is subtractive. Let x AM R. Then x = a b for some a, b A. i.e., a = x∈+ b =∩ x A. − ∈ ⇒ ∈ (b) If A B, then it is clear that AM = a b a, b A ⊆ { − | ∈ } v a b a, b B = BM. { − | ∈ } 2. The result is simple and straight forward. 3. (a) Assume that AM BM. Then A = AM R BM R = B, as A and B are subtractive.⊆ ∩ ⊆ ∩ (b) A = B = AM = BM. Hence AM ( BM = A ( B. ⇒ ⇒ 4.( P R)M P is obvious. Now let x = a b P . Since R is ∩ ⊆ − ∈ assumed to be yoked, for all a b RM, either a b R or b a R. − ∈ − ∈ − ∈ Therefore either a b P R or b a P R = a b (P R)M. − ∈ ∩ − ∈ ∩ ⇒ − ∈ ∩

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5. If P Q, then clearly P R Q R. ⊆ ∩ ⊆ ∩

Theorem 14. Let R be an additively cancellative sub-distributive semiring and R∆ its sub-distributive ring of differences.

1. If A is a subtractive prime ideal of R where R is yoked, then AM is a prime ideal of R∆; 2. If P is a prime ideal of R∆, then P R is a prime ideal of R. ∩ Proof. Given R an additively cancellative sub-distributive semir- ing and R∆ its sub-distributive ring of differences 1. Suppose I,J be two ideals of R∆ such that IJ A∆. Using Lemma 12 we get ⊆

(I R)(J R) (IJ) R AM R = A. ∩ ∩ ⊆ ∩ ⊆ ∩ This implies either I R A or J R A, since A is prime. ∆∩ ⊆ ∩ ⊆∆ This implies (I R) AM or (J R) AM. Since R is ∩ ⊆ ∩ ⊆ yoked, we have I AM or J AM by using lemma 12(4). ⊆ ⊆ 2. Let A, B be two ideals of R such that AB P R. This ⊆ ∩ implies AMBM = (AB)M (P R)M P. Since P is prime, ⊆ ∩ ⊆ either AM P or BM P. ⊆ ⊆ Again by Lemma 12(1a), A AM R P R or B ⊆ ∩ ⊆ ∩ ⊆ BM R P R. Hence P R is a prime ideal of R. ∩ ⊆ ∩ ∩

References

[1] J. S. Golan, Semirings and their applications, Kluwer Aca- demic Publishers, Dordrecht, Boston, London, 1999. [2] M. Lorenz and D. S. Passman, Prime ideals in crossed products of finite groups, Israel J. Math. 33 (1979), 89-132. [3] Ram Parkash Sharma and Rosy Joseph, Prime Ideals of Group Graded Semirings and Their Smash Products, Vietnam Jour- nal of Mathematics 36:4 (2008) 415-426.

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[4] Rosy Joseph and Dhanalakshmi V, Structure study of Sym- metric Fuzzy Numbers, Presented at the Bilateral Interna- tional Conference FIM 2017 & ISME2017, Twenty-Sixth Inter- national Conference Forum of Interdisciplinary Mathematics FIM2017 & International Symposium on Management Engi- neering ISME 2017 at Kitakyushu, Fukuoka, Japan (2017).

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