Journal of Computer and Mathematical Sciences, Vol.7(5), 294-301, May 2016 ISSN 0976-5727 (Print) (An International Research Journal), www.compmath-journal.org ISSN 2319-8133 (Online)
Structure of Some Idempotent Semirings
N. Sulochana1 and T. Vasanthi2
1Assistant Professor, Department of Mathematics, K.S.R.M College of Engineering, Kadapa-516 003, A. P., INDIA. Email: [email protected] 2Professor of Mathematics, Department of Mathematics, Yogi Vemana University, Kadapa-516 003, A. P., INDIA.
(Received on: May 30, 2016)
ABSTRACT
In this paper we study the notion of some special elements such as additive idempotent, idempotent, multiplicatively sub-idempotent, regular, absorbing and characterize these subidempotent and almost idempotent semiring. It is prove that, if S is a subidempotent semiring and (S, ) be a band, then S is a mono semiring and S is k - regular.
2000 Mathematics Subject Classification: 20M10, 16Y60.
Keywords: Integral Multiple Property (IMP); Band; Mono Semirings; Rectangular Band; Multiplicatively Subidempotent; k-regular.
1. INTRODUCTION
The concept of semirings was first introduced by Vandiver in 1934. However the developments of the theory in semirings have been taking place since 1950. The theory of rings and the theory of semigroups have considerable impact on the developments of the theory of semirings and ordered semirings. It is well known that if the multiplicative structure of an ordered semiring is a rectangular band then its additive structure is a band. P.H. Karvellas2 has proved that if the multiplicative structure of a semiring is a regular semigroup and if the additive structure is an inverse semigroup, then the additive structure is commutative. There is considerable impact of semigroup theory and semiring theory on the development of ordered semirings both in theory and applications, which are Akin to ordered rings and ordered semirings. In this direction the works of H. J. Weinert6, M. Satyanarayana, J.Hanumanthachari and D. Umamaheswara Reddy4, They tend to arise whenever we use dynamic programming in
294 N. Sulochana, et al., Comp. & Math. Sci. Vol.7 (5), 294-301 (2016) order to find the "best" of some set of things: e.g., the most likely tag sequence, the shortest path, the longest path, etc. This is what part of speech tagging, matrix chain multiplication, and Dijkstra's algorithm are all about. The concept of subidempotent semiring is taken from the book of Jonathan S. Golan1, entitled “Semirings and their Applications”.
2. PRELIMINARY DEFINITIONS
Definition 2.1: A triple (S, +, •) is said to be a semiring if S is a non - empty set and “+, •” are binary operations on S satisfying that (i) (S, +) is a semigroup (ii) (S, •) is a semigroup (iii) a(b + c) = ab + ac and (b + c)a = ba + ca, for all a, b, c in S.
Examples of Semiring
(i) The set of natural numbers under the usual addition, multiplication (ii) Every distributive lattice (L, ∨, ∧ ). (iii) Any ring (R, +, •).
Definition 2.2: An element a of a semiring S is multiplicatively subidempotent if and only if a + a2 = a and a semiring S is multiplicatively subidempotent if and only if each of its elements is multiplicatively subidempotent.
Definition 2.3: 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.
Definition 2.4: An additive idempotent semiring S is k – regular if for all a in S there is a b in S for which a + aba = aba.
Definition 2.5: A semiring S is said to be mono semiring if a + b = ab for all a, b in S.
Definition2.6: An element a of a semigroup (S, ·) is left (right) regular if there exists an element b in S such that ba2 = a (a2b = a). A semigroup (S, ·) is called left (right) regular if every element of S is left (right) regular.
Definition 2.7: (i) A semigroup (S, +) satisfying the identity a = a + b + a is a rectangular band.
(ii) A semigroup (S, ·) satisfying the identity a = aba is a rectangular band.
3. STRUCTURE OF SUBIDEMPOTENT SEMIRINGS
In this section we studied some properties of subidempotent semirings by considering different conditions.
Definition 3.1: A semigroup (S, ·) is called left (right) permutable if for every a, b, c in S, abc = acb (abc = bac). A semigroup S is called permutable if it is both left and right permutable.
295 N. Sulochana, et al., Comp. & Math. Sci. Vol.7 (5), 294-301 (2016)
Definition 3.2: A C – semiring is a semiring in which
(i) (S, +) is a commutative monoid (ii) (S, ·) is a commutative monoid (iii) a (b + c) = ab + ac and (b + c) a = ba + ca for every a, b, c in S (iv) a.0 = 0.a = 0 (v) (S, +) is a band and 1 is the absorbing element of ‘+’.
Definition 3.3: A semiring is said to be subidempotent semiring if S satisfies the identity a + ab = b for every a, b S.
Theorem 3.4: Let S is a subidempotent semiring. (i) If (S, ·) is left permutable then (S, ·) commutative. (ii) Suppose (S, ·) is a right regular semigroup then (S, ·) is left regular semigroup
Proof: (i) To show that (S, ·) is commutative Consider a + ab = b (I) a2 + aba = ba Since (S, ·) is left permutable we have aba = aab a2 + aab = ba a ( a + ab ) = ba On substitution of condition (I) we get ab = ba Therefore (S, ·) is commutative. (ii) Since S is a subidempotent semiring a + ab = b (I) Since (S, ·) is right regular we have a = a2b a2b + (a2b) b = b a (a + ab)b = b By condition (I) it takes the form a(b)b = b ab2 = b Hence (S, ·) is left regular semigroup.
Theorem 3.5: If S is a subidempotent semiring and b + a = b then b + an = b
Proof: By the definition of subidempotent we have a + ab = b (I) Using the condition b + a = b in above it reduces to a + a(b +a) = b a + ab+ a2 = b Using first equation we get b + a2 = b ab + a3 = ab On addition of a we have a + ab + a3 = a + ab Which implies b + a3 = b Continuing in a similar way, we get b + an = b
Theorem 3.6: Let S be a subidempotent semiring and (S, ) be a band. Then the following hold (i) S is a mono semiring. (ii) S is k - regular.
296 N. Sulochana, et al., Comp. & Math. Sci. Vol.7 (5), 294-301 (2016)
Proof: (i) Given that S is a subidempotent semiring thus we have a + ab = b Using the condition that (S, ) is a band, a2 = a then above equation is written as a2 + a2b = b a (a + ab) = b Since a + ab = b it takes the form ab = b ------(I) Again a + ab = b is reduced to a + b = b ------(II) Therefore from equations (I) and (II) we conclude that a + b = ab Hence S is a mono semiring (ii) Consider aba = a (a + ab) a = a (a2 + aba) = a (a + aba) = a2 + a2ba Since (S, ) is a band we get aba = a + aba Thus we proved that S is k – regular
Example: Here we framed an example by considering a three element set S = {a, b, c} which satisfies the conditions of the above theorem + a b c · a b c a a b c a a b c b a b c b a b c c a b c c a b c
Theorem 3.7: If S is a subidempotent semiring and (S, ·) is a rectangular band, then (i) S is multiplicatively subidempotent and (ii) If (S, +) is left cancellative, then anb = ab for all a, b in S.
Proof: (i) Suppose (S, ·) is rectangular band, aba = a for all a, b in S Given that S is a subidempotent semiring a + ab = b a2 + aba = ba a2 + a = ba a3 + a2 = aba a + a2 = a ------(I) Again a + ab = b ba + bab = b2 ba + b = b2 bab+ b2 = b3 b + b2 = b------(II) Hence S is multiplicatively subidempotent
(ii) We can still show that anb = ab for all a, b in S For this consider a + ab = b a2 + a2b = ab a + a2 + a2b = a + ab From (i), we have a + a2 = a, which implies a + a2b = a + ab a + a2b = b a2 + a3b = ab a + a2 + a3b = a + ab a + a3b = a + ab a + a3b = b Continuing like this a + anb = b for all n > 0 And also a + anb = a + ab By using left cancellation we get anb = ab, for n > 0 Hence the theorem
297 N. Sulochana, et al., Comp. & Math. Sci. Vol.7 (5), 294-301 (2016)
Theorem 3.8: If S is a subidempotent semiring and (S, +) is commutative, then b + a + b = b is true in the following cases (i) a + b + ab = ab (ii) a + b + a = a
Proof: (i) First assume that a + b + ab = ab a + b + ab + a = ab + a Since (S, +) is commutative we get a + b + a + ab = a + ab By the hypothesis a + ab = b thus a + b + b = b b + a + b = b (ii) Assume that a + b + a = a a + b + a + ab = a + ab a + b + b = b Using (S, +) is commutative in above equation reduces to b + a + b = b Thus we proved b + a + b = b
Theorem 3.9: If S is a C-semiring which is also subidempotent semiring then an + b = b
Proof: By hypothesis S is a C-semiring, in which (S, +) is an absorbing, a + 1 = 1 ab + b = b a + ab + b = a + b b + b = a + b Since (S, +) is band, b + b = b Therefore a + b = b ---- (I) a2 + ab = ab a + a2 + ab = a + ab a2 + a + ab = b a2 + b = b -----(II) Again a3 + ab = ab a + a3 + ab = a + ab a3 + a + ab = b a3 + b = b -----(III) Continuing in this manner, we can conclude that an + b = b
4. STRUCTURE OF ALMOST IDEMPOTENT SEMIRINGS
Definition 4.1: An element a of a semiring S is called an almost idempotent if a + a2 = a2. A semiring is said to be almost idempotent if every element in S is almost idempotent. We denote the set of all almost idempotents of a semiring S by Ek (S).
Definition 4.2: A semiring (S, +, ·) is said to satisfy the Integral Multiple Property (IMP) if a2 = na for all a in S where the positive integer n depends on the element a.
Definition 4.3: An element a is periodic if ma = na where m and n are positive integers. A semigroup S is said to be periodic if every one of its elements is periodic.
Definition 4.4: Zeroid of a semiring (S, +, ·) is the set of all a in S such that a + b = b or b + a = b for some b in S, we may also this term as the zeroid of (S, +).
298 N. Sulochana, et al., Comp. & Math. Sci. Vol.7 (5), 294-301 (2016)
Definition 4.5: (i) A semigroup (S, ·) is said to be a band if every element in S is an idempotent. (ii) A commutative band is called a Semilattice.
Theorem 4.6: Suppose S is a semiring and a Ek (S). If S satisfies IMP then a is periodic element with respect to ‘+’.
2 2 Proof: First let us take a Ek(S). This implies a + a = a ----(I) By hypothesis, S satisfies IMP, a2 = na From above two equations we get, a + na = na (n + 1) a = na Thus a is periodic element with respect to addition
Theorem 4.7: Let S be a semiring and a, b Ek(S). If S contains multiplicative identity (i) If (S, ·) is left cancellative, then a + b + ab = ab. (ii) If a + b = 1 then a, b E (+).
2 2 2 2 Proof: (i). Let us consider a, b Ek (S) implies a + a = a and b + b = b a (1 + a) = a.a and b (1 + b) = b.b (1 + a) = a and (1 + b) = b Let us take a + b + ab = a + (1 + a) b = a + ab = a (1 + b) = ab Hence a + b + ab = ab (ii). Suppose if a + b = 1 implies a2 + ab = a a + a2 + ab = a + a Since we have S is almost idempotent implies a2 + ab = a + a a (a + b) = a + a a.1 = a + a Thus a = a + a Similarly we can prove b = b + b Hence a, b E(+)
Theorem 4.8: If S is almost idempotent semiring, and (S, +) is right cancellative, then (S, +) is a band if and only if (S, ·) is band.
Proof : Suppose (S, +) is a band a + a = a a2 + a2 = a2 ----(I) 2 2 Also a Ek (S) which implies a + a = a ---- (II) Equating (I) and (II), we obtain a + a2 = a2 + a2 Using (S, +) is left cancellation, a = a2 By retracing the steps, we obtain the converse
Theorem 4.9: let S be an almost idempotent semiring.
(i) If S is viterbi semiring then S is idempotent semiring. (ii) If (S, +) is zeroid and right cancellative then (S, ·) is semilattice and (S, ·) regular.
299 N. Sulochana, et al., Comp. & Math. Sci. Vol.7 (5), 294-301 (2016)
Proof: (i) Assume that S is a viterbi semiring a + a2 = a and a + a = a Suppose S is almost idempotent semiring a + a2 = a2 a = a2 and a + a = a From above equations we can conclude that S is idempotent semiring (ii) Consider (S, +) is zeroid we have a + b = b a2 + ab = ab a + a2 + ab = a + ab a2 + ab = a + ab By using right cancellation it reduces to the form a2 = a Therefore (S, ·) is band. Let us take the almost idempotent semiring a + a2 = a2 a + a2 + ab = a2 + ab a + a (a + b) = a (a + b) a + ab = ab a + ab + b2 = ab + b2 a + (a + b) b = (a + b) b a + b2 = b2 ------(I) Again a + b = b ab + b2 = b2 and ba + b2 = b2 ------(II) From (I) and (II) we have a + b2 = ab + b2 = ba + b2 a = ab and ba = a Therefore (S, ·) is commutative Hence (S, ·) is semilattice. implies a = ab a2 = aba a = aba Thus (S, ·) regular
Theorem 4.10: Let S be an almost idempotent and mono semiring. Then (S, +) is k- regular
Proof: Since S is a mono semiring we have a + b = ab ----(I) a2 + ba = aba a + a2 + ba = a + aba a2 + ba = a + aba (a + b) a = a + aba Using equation (I) we obtain aba = a + aba Thus (S, +) is k- regular
Theorem 4.11: Let S be a semiring with multiplicative identity 1 satisfying the identity a + b = 1 for all a, b S. If (S, +) is rectangular band, then S is almost idempotent semiring.
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Proof: a + b = 1 a2 + ab = a a2 + ab + a2 = a + a2 a(a + b + a) = a + a2 a + a2 = a2 Again a + b = 1 ab + b2 = b b2 + ab + b2 = b2 + b (b + a + b) b = b2 + b b2 = b2 + b Hence S is almost idempotent semiring
REFERENCES
1. Jonathan S. Golan, “Semrings and Their Applications”, Kluwer Academic Publishers, Dordrecht, (1999). 2. P. H. Karvellas “Inversive semirings”. Jour. Aust. Math. Soc., Vol.18, 277-287 (1974). 3. M. Satyanarayana – “On the additive semigroup of ordered Semirings”, Semigroup Forum Vol.31 (1985), 193-199. 4. M. Satyanarayana, J. Hanumanthachari and D. Umamaheswara Reddy “On the additive structure of a class of ordered semirings”. Semigroup Forum Vol.33, 251-255 (1986). 5. T. Vasanthi & N. Sulochana, Semirings Satisfying The Identities, International Journal of Mathematical Archive – 3 (9), 3393-3399 Sept. – (2012). 6. H.J.Weinert :“Multiplicative cancellativity of semirings and semigroups”. Acta. Mathematica. Academicae, Scientiarum Hungaricae, Tomus, 35 (3-4), 335-338 (1980).
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