Some Studies on Simple Semiring

Some Studies on Simple Semiring

1. Sreenivasulu Reddy and Abduselam Mahamed Derdar

   Department of mathematics, Samara University Semera, Afar Regional State, Ethiopia. Post Box No.131

   Abstract: Authors determine different additive and multiplicative structures of simple semiring which was introduced by Golan [1]. We also proved some results based on the paper P. Sreenivasulu Reddy and Guesh Yfter tela [4].

   1. INTRODUCTION

This paper reveals the properties of simple semiring. Through out this paper simple semiring (S, +, ) means simple semiring (S, +, ) with multiplicative identity 1.

1. Definition: A triple (S, +, ) is said to be a semiring if S is a non – empty set and +, are binary operations on S satisfying that

   1. (S, +) is a semigroup

   2. (S, ) is a semigroup

   3. a(b + c) = ab + ac and (b + c)a = ba + ca, for all a, b, c in S.

      Examples: (i) The set of natural numbers under the usual addition, multiplication

      1. Every distributive lattice (L, , ).

      2. Any ring (R, +, ).

      3. If (M, +) is a commutative monoid with identity element zero then the set End(M) of all endomorphism of M is a semiring under the operations of point wise addition and composition of functions.

   (vi) Let S = {a, b} with the operations given by the following tables:

   +	a	b
   a	a	b
   b	b	b

   	a	b
   a	b	b
   b	b	b

   Then (S, +, ) is a semiring.

2. Definition: An element x in a semigroup (S, ) is said to be multiplicative idempotent if x2 = x. 1.3.Definition: An element x in a semigroup (S, +) is said to be an additive idempotent if x + x = x.

1. Definition: A semigroup (S, ) with all of its elements are left (right) cancellable is said to be left (right) cancellative semigroup.

2. Definition: A semigroup (S, ) is said to satisfy quasi separative if x2 = xy = yx = y2 x = y, for all x, y in S.

3. Definition: A semigroup (S, +) is said to satisfy weakly separative if x + x = x + y = y + y x = y, for all x, y in S.

4. Definition: A semigroup (S, ) is said to be left (right) regular if it satisfies the identity aba = ab (aba = ba) for all a, b in S.

5. Definition: A semigroup (S, +) is said to be left (right) singular if it satisfies the identity a + b = a (a + b = b) for all a, b in S.

6. Definition: A semigroup (S, .) is said to be left(right) singular if it satisfies the identity ab = a (ab = b) for all a,b in S

7. Definition: [3] A semiring S is called simple if a + 1 = 1 + a = 1 for any aS.

8. Definition: A semiring (S, +, .) with additive identity zero is said to be zero sum free semiring if x + x = 0 for all x in S.

9. Definition: . A semiring (S, +, . ) is said to be zero square semiring if x2 = 0 for all x in S, where 0 is multiplicative zero.

10. Definition: A viterbi semiring is a semiring in which S is additively idempotent and multiplicatively subidempotent.i.e., a

    + a = a and a + a2 = a, for all a in S.

11. Theorem: A simple semiring is additive idempotent semiring.

    Proof: Let (S, +, .) be a simple semiring. Since (S, +, .) is simple, for any aS, a + 1 = 1. (Where 1 is the multiplicative identity element of S. S1 = SU {1}.)

    Now a = a.1 = a(1 + 1) = a + a a = a + a S is additive idempotent semiring.

12. Theorem: If (S, +, ) be a simple semiring and (S, +) be a right cancellative then (S, ) be a band.

    Proof: From hypothesis, (S, +, ) be a simple semiring a + 1 = 1 a(a + 1) = a.1 a2 + a = a a2 + a = a + a (Since Theorem1.14) a2 = a (Since (S, +) be a right cancellative) (S, ) be a band.

13. Theorem: If (S, +, ) be a simple semiring and (S, ) be a rectangular band then (S, ) be a singular.

    Proof: From hypothesis, (S, +, ) be a simple semiring a + 1 = 1 b(a + 1) = b.1 ba + b = b a(ba + b) = ab aba + ab

    = ab a + ab = ab (Since (S, ) be a rectangular band) a(1 + b) = ab a = ab ab= a (S, ) be a left singular. (1) Again, a + 1 = 1 (a+1)b = 1.b ab + b = b (ab + b)a = ba aba + ba = ba a + ba = ba (Since (S, ) be a rectangular band) a(1+b)=ba a = ba ba = a (S, ) be a right singular. (2)

    From (1) and (2), (S, ) be a singular.

14. Theorem: If (S, +, ) be a zero sum free and simple semiring with additive identity 0 then ab = 0 for every a, b in (S, +, ). Proof: Since (S, +, ) be a simple semiring, b + 1 = 1 a(b + 1)=a.1 ab + a = a ab + a + a = a + a (Since theorem1.14) ab + 0 = 0 ((S, +, ) be a zero sum free semiring) ab = 0.

15. Theorem: If (S, +, ) be a zero square and simple semiring with additive identity 0 then aba = 0 and bab = 0 for every a, b in (S, +, ).

    Proof: Since (S, +, ) be a simple semiring, b + 1 = 1 a(b + 1)=a.1 ab + a = a (ab + a)a = a.a aba + a2 = a2 aba + 0

    = 0 (Since (S, +, ) be a zero square semiring) aba = 0.

    Again, a + 1 = 1 b(a + 1)=b.1 ba + b = b (ba + b)b = b.b bab + b2 = b2 bab + 0 = 0 (Since (S, +, ) be a zero square semiring) bab = 0.

16. Theorem: Let (S, +, ) be a simple semiring. If (S, ) is a singular then (S, +) is a singular.

    Proof: Let (S, +, ) be a simple semiring in which (S, ) is a singular that is ab = a ab + b = a + b (a + 1)b = a + b b = a

    + b a + b = b (S, +) is a right singular. (1)

    Again, ab = b a + ab = a + b a(1 + b) = a + b a.1 = a + b a = a + b a + b = a (S, +) is a left singular. (2) From (1) and (2), (S, +) is a singular.

    Example: The following example satisfies the conditions of theorem

    +	1	a	b
    1	1	a	b
    a	1	a	b
    b	1	a	b

    .	1	a	b
    1	1	a	b
    a	a	a	a
    b	b	a	b

17. Theorem: Let (S, +, ) be a simple semiring. If (S, ) be a left regular semigroup then (S, +) is an E-inversive semigroup E(+).

    Proof: By hypothesis (S, ) be a left regular semigroup then aba = ab for every a, b in (S, )

    b+ 1 = 1 a(b + 1) = a.1 ab + a = a b(ab + a) = ba bab + ba = ba ba + ba = ba, a,bE(+). Where E(+) is the set of all idempotent elements in (S, +). This means that there exists a in S such that ba + ba = ba implies ba is an E-inversive element. Hence (S, +) is an E-inversive semigroup.

18. Theorem: If (S, +, ) be a simple semiring with multiplicative identity which is also additive identity then (S, ) is a quasi- seperative semigroup.

    Proof: If (S, +, ) be a simple semiring with multiplicative identity which is also additive identity then ab + a = a. Let a2 = ab a2 = a(b + e) a2 = ab + a.e a2 = ab +a a2 = a.

    Similarly, b2 = ba b2 = b(a + e) b2 = ba + b.e b2 = ba + b b2 = b. If a2 = ab = ba = b2 then a = b. Hence (S, ) is a quasi-seperative semigroup.

19. Theorem: If (S, +, ) be a simple semiring with multiplicative identity which is also additive identity then (S, ) is a (i) seperative semigroup.

    (ii) weakly seperative semigroup.

    Proof: Proof is similar to above theorem1.22.

20. Theorem: Every simple semiring (S, +, ) is a viterbi semiring. Proof: By hypothesis (S, +, ) be a simple semiring

    From the theorem1.14 (S, +, ) be an additive idempotent semiring that is a + a = a (1) And 1 + a = 1 a(1 + a) = a a + a2 = a (2)

    From (1) & (2), (S, +, ) is a viterbi semiring.

    Remark: Converse of theorem1.15, is true if (S, ) is left cancellative and (S, +) is commutative. Proof: Consider a + a2 = a, for all a in S

    a.1 + a2 = a .1

    a (1 + a) = a .1

    1 + a = 1 (Since (S, ) is left cancellative)

    1 + a = a + 1 = 1 (Since (S, +) is commutative)

    (S, +, ) be a simple semiring.

    Example: This is an example for theorem 1.23

    +	1	a
    1	1	1
    a	1	a

    	1	a
    1	1	a
    a	a	a

21. Theorem: Every simple semiring (S, +, ) is a multiplicative sub idempotent semiring. Proof: Proof is similar to above theorem1.23.

REFERENCES:

1. Golan, J.S. The theory of semirings with applications in mathematics and theoretical computer science, Pitman monographs and surveys in pure and applied mathematics, II. Series.(1992).

2. Jonathan S.Golan, Semirings and their Applications.

3. Jonathan S.Golan, Semirings and Affine Equations over Them : Theory and Applications. Kluwer Academic.

4. P. Sreenivasulu Reddy and Guesh Yfter tela Simple semirings, International journal of Engineering Inventions, Vol.2, Issue 7, 2013, PP: 16-19.