BOOLEAN LIKE SEMIRINGS C. Venkata Lakshmi #1, T. Vasanthi #2 #1 Dept. of Applied Mathematics, Sri Padmavati Women s University, Tirupati, A.P., India. #2 Dept. of Applied Mathematics, Yogi Vemana University, Kadapa 516003, A.P., India. ABSTRACT In this paper, we discuss bolean like semirings. This paper deals with some definitions, properties of bolean like semirings and ordered bolean like semirings. Here we proved that in a boolean like semiring (S, +, ), if (S, +) is regular, then (S, ) is a band. Key words:band; Left (Right) singular; Regular; Multiplicatively subidempotent; zero square elements ; Multiplicative ideal; E - inverse semigroup; Minimum (Maximum) element; Positively totally ordered semigroup; Negatively totally ordered semigroup. 2000 Mathematics Subject Classification: 20M10, 16Y60. Corresponding Author: C. Venkata Lakshmi INTRODUCTION Semirings abound in the mathematical world around us. A semiring is one of the fundamental structures in mathematics. Indeed the first mathematical structure we encounter the set of natural numbers is a semiring. Other semirings arise naturally in such diverse areas of mathematics as combinatorics, functional analysis, topology, graph theory, Euclidean geometry, probability theory, commutative, non-commutative ring theory and the mathematical modeling of quantum physics and parallel computation systems. The modern interest in semirings arises primarily from fields of Applied Mathematics such as Optimization theory, the theory of discrete-event dynamical systems, automata theory and formal language theory, as well as from the allied areas of theoretical computer science and theoretical physics and the questions being asked is, for the most part, motivated by applications. Semirings have been studied by various researchers [1,2,3,4,5] in an attempt to broaden techniques coming from the semigroup theory or ring theory or in connection with applications. The semiring identities are taken from the book of Jonathan S.Golan[2], entitled Semirings and their Applications. In this paper we investigate the additive and multiplicative properties of Boolean Like Semirings. R S. Publication (rspublication.com), rspublicationhouse@gmail.com Page 316
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. A semiring (S, +, ) is said to be totally ordered semiring (t.o.s.r.) if there exists a partially order on S such that (i) (S, +) is a t. o. s. g. (ii) (S, ) is a t. o. s. g. It is usually denoted by (S, +,, ). In a totally ordered semiring (S, +,, ) (i) (S, +, ) is positively totally ordered (p.t.o.), if a + b a, b for all a, b in S and (ii) (S,, ) is positively totally ordered (p.t.o.), if ab a, b for all a, b in S. In a totally ordered semiring (S, +,, ) (i) (S, +, ) is negatively totally ordered (n.t.o.),if a + b a, b for all a, b in S and (ii) (S,, ) is negatively totally ordered (n.t.o.), if ab a, b for all a, b in S. 1. BOOLEAN LIKE SEMIRINGS In this section, the properties of boolean like semirings are studied. We proved that if (S, +, ) is a boolean like semiring, then the set X of all zero square elements is a multiplicative ideal of S. 1.1 A non empty set S together with two binary operations ` + and ` satisfying the following conditions is called a boolean like semiring (i) (S, +) is a semigroup (ii) (S, ) is a semigroup (iii) a.(b + c) = a.b + a.c and (b + c).a = b.a + c.a (iv) ab (a + b + ab) = ab, for all a,b in S and a.0 = 0.a = 0 (v) Weak commutative: a.b.c = b.a.c, for all a, b, c in S 1.2 An element `a of a semiring S is multiplicatively subidempotent if and only if a + a 2 = a and S is multiplicatively subidempotent if and only if each of its elements is multiplicatively subidempotent. THEOREM 1.3:Let (S, +, ) be a boolean like semiring. If S contains the multiplicative identity 1 and a + 1 = 1, then (S, ) is multiplicatively subidempotent. R S. Publication (rspublication.com), rspublicationhouse@gmail.com Page 317
PROOF:Given that (S, +, ) is a boolean like semiring We have ab (a + b + ab) = ab, for all a, b in S Taking b = 1 a (a + 1 + a) = a a (1 + a) = a ( a + 1 = 1) a + a 2 = a (S, ) is multiplicatively subidempotent. 1.4 A semigroup (S, +) is said to be regular if it satisfying the identity x = x + y + x, for every x in S. THEOREM 1.5:Let (S, +, ) be a boolean like semiring containing the multiplicative identity 1. If (S, +) is regular, then (S, ) is a band. PROOF:Considerab (a + b + ab) = ab, for all a, b in S Taking b = 1 a(a + 1 + a.1) = a.1 a(a + 1 + a) = a a(a) = a a 2 = a ( (S, +) is regular, a + 1+ a = a) (S, ) is a band. 1.6 Anelement `a of a multiplicative semigroup `S is called an E inverse if there is an element `x in S such that ax + ax = ax, i.e. ax E (+), where E (+) is the set of all multiplicative idempotent elements of S. A Semigroup `S is called an E inverse Semigroup if every element of S is an E inverse. 1.7 Anelement `a of a multiplicative semigroup `S is called an E inverse if there is an element `x in S such that ax (ax) = ax, i.e. ax E ( ), where E ( ) is the set of all multiplicative idempotent elements of S. A Semigroup `S is called an E inverse Semigroup if every element of S is an E inverse. THEOREM 1.8:Let (S, +, ) be a boolean like semiring containing the multiplicative identity which is also an additive identity, then (S, ) is an E - inverse semigroup. PROOF:By hypothesis,(s, +, ) is a boolean like semiring Let `e be the multiplicative identity is also an additive identity Consider ab (a + b + ab) = ab, for all a, b in S ab (a + (e + a)b) = ab ab (a + ab) = ab ab (a (e + b)) = ab ab (ab) = ab R S. Publication (rspublication.com), rspublicationhouse@gmail.com Page 318
ab E[ ] Hence (S, ) is an E - inverse semigroup. 1.9 An element x in a semigroup (S, +) is said to be an absorbing if x + a = a + x = x, for every a in S. THEOREM 1.10:Let (S, +, ) be a boolean like semiring containing the multiplicative identity 1, which is also an absorbing element w.r.to. `+. If (S, ) is left cancellative, then a + b = 1, for all a, b in S. PROOF:Since (S, +, ) is a boolean like semiring Let 1 be the multiplicative identity is also an absorbing element w.r.to.`+ Consider ab (a + b + ab) = ab, for all a, b in S ab (a + (1 + a) b) = ab.1 ab (a + b) = ab.1 ( 1 + a = 1) a + b = 1 ( (S, ) is left cancellative) a + b = 1, for all a, b in S. 1.11 A semigroup (S, ) is said to be (i) left singular, if it satisfies the identity ab = a, for all a, b in S; (ii) right singular, if it satisfies the identity ab = b, for all a, b in S; (iii) singular, if it is both left as well as right singular. 1.12 A semigroup (S, +) is said to be (i) left singular, if it satisfies the identity a + b = a, for all a, b in S; (ii) right singular, if it satisfies the identity a + b = b, for all a, b in S; (iii) singular, if it is both left as well as right singular. 1.13 A semigroup (S, ) is said to be (i) left regular, if it satisfies the identity aba = ab, for all a, b in S; (ii) right regular, if it satisfies the identity aba = ba, for all a, b in S; (iii) regular, if it is both left as well as right regular. THEOREM 1.14:Let (S, +, ) be a boolean like semiring containing the multiplicative identity 1. If (S, +) is left singular, then (S, ) is left regular. PROOF:Considerab (a + b + ab) = ab, for all a, b in S ab (a + a b) = ab ( (S, +) is a left singular, a + b = a) ab (a [1 + b] ) = ab ab (a.1) = ab ( (S, +) is a left singular, 1 + b = 1) aba = ab (S, ) is left regular. R S. Publication (rspublication.com), rspublicationhouse@gmail.com Page 319
1.15 In a semigroup (S, ), a non-empty subset A of S is called (i) a multiplicative left ideal, if sa A, for every s S and for every a A (ii) a multiplicative right ideal, if as A, for every a A and for every s S (iii) a multiplicative ideal, if A is both a left ideal as well as a right ideal 1.16 A semiring (S, +, ) with multiplicative zero is said to be zero - square semiring if x 2 = 0, for all x S. 1.17 A semiring (S, +, ) with additive identity zero is said to be zerosumfree semiring if x + x = 0, for all x S. THEOREM 1.18:Let (S, +, ) be a boolean like semiring. Then the set X of all zero square elements is a multiplicative ideal of S. PROOF:Let s S and x X (xs) 2 = xs. xs (xs) 2 = sxxs ( by weak commutative : xsx = sxx) (xs) 2 = sx 2 s (xs) 2 = x 2 ss ( by weak commutative) (xs) 2 = x 2 s 2 (xs) 2 = 0.s 2 = 0 ( x is a zero square element, x 2 = 0) xs X Similarly, (sx) 2 = sx. sx (sx) 2 = xs sx ( by weak commutative: sxs = xss) (sx) 2 = xs 2 x (sx) 2 = s 2 x x ( by weak commutative) (sx) 2 = s 2 x 2 (sx) 2 = s 2. 0 = 0 ( x is a zero square element, x 2 = 0) sx X Hence X is a multiplicative ideal. 1.19 An element x in a totally ordered semiring (t.o.s.r) is minimal (maximal) if x a (x a) for every a in S. THEOREM 1.20:Let (S, +, ) be a totally ordered boolean like semiring. If (S, ) is positively totally ordered (p.t.o.) [negatively totally ordered (n.t.o.)], then 0 is the maximum (minimum) element. PROOF:Since (S, +, ) is a totally ordered boolean like semiring We have a.0 = 0.a = 0, for all `a in S Suppose (S, ) is positively totally ordered (p.t.o.) Then a.0 a and 0 a.0 a 0 a ( a.0 = 0) R S. Publication (rspublication.com), rspublicationhouse@gmail.com Page 320
0 is the maximum element. Suppose (S, ) is negatively totally ordered (n.t.o.) Then a.0 a and 0 a.0 a 0 a ( a.0 = 0) 0 is the minimum element. CONCLUSION 1. In Boolean Like Semirings, the algebraic structure of a multiplicative semigroup (S, ) determine the additive structure of (S, +) and viceversa. 2. In a Boolean like semiring, if the multiplicative semigroup (S, ) is positively totally ordered (negatively totally ordered) then 0(zero) becomes the maximum (minimum) element. REFERENCES 1. Arif Kaya and M.Satyanarayana Semirings satisfying properties of distributive type, Proceeding of the American Mathematical Society, Volume 82, Number 3, July 1981 2. Jonathan S.Golan Semirings and their Applications. 3. Jonathan S.Golan Semirings and Affine Equations over Them : Theory and Applications. Kluwer Academic Publishers (1999). 4. M.Satyanarayana On the additive semigroup of ordered semirings,semigroup forum vol.31 (1985), 193-199 5. M.P.Grillet Subdivision rings of a semiring. Fund. Math., Vol.67 (1970), 67-74. R S. Publication (rspublication.com), rspublicationhouse@gmail.com Page 321