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Relations on Semigroups International Journal for Research in Engineering Application & Management (IJREAM) ISSN : 2454-9150 Vol-04, Issue-09, Dec 2018 Relations on Semigroups 1D.D.Padma Priya, 2G.Shobhalatha, 3U.Nagireddy, 4R.Bhuvana Vijaya 1 Sr.Assistant Professor, Department of Mathematics, New Horizon College Of Engineering, Bangalore, India, Research scholar, Department of Mathematics, JNTUA- Anantapuram [email protected] 2Professor, Department of Mathematics, SKU-Anantapuram, India, [email protected] 3Assistant Professor, Rayalaseema University, Kurnool, India, [email protected] 4Associate Professor, Department of Mathematics, JNTUA- Anantapuram, India, [email protected] Abstract: Equivalence relations play a vital role in the study of quotient structures of different algebraic structures. Semigroups being one of the algebraic structures are sets with associative binary operation defined on them. Semigroup theory is one of such subject to determine and analyze equivalence relations in the sense that it could be easily understood. This paper contains the quotient structures of semigroups by extending equivalence relations as congruences. We define different types of relations on the semigroups and prove they are equivalence, partial order, congruence or weakly separative congruence relations. Keywords: Semigroup, binary relation, Equivalence and congruence relations. I. INTRODUCTION [1,2,3 and 4] Algebraic structures play a prominent role in mathematics with wide range of applications in science and engineering. A semigroup is one of the algebraic structure, a set with one binary operation satisfying the law of associativity. The binary operation of a semigroup is most often denoted multiplicatively: x·y, or simply xy, denotes the result of applying the semigroup operation to the ordered pair(x, y). Associativity is formally expressed as that (x·y)·z = x·(y·z) for all x, y and z in the semigroup A semigroup generalizes a group by preserving only associativity and closure under the binary operation from the axioms defining a group while omitting the requirement for an identity element and inverses. The theory of semigroups is one of the relatively young branch of algebra.These algebraic structures are important in many areas of mathematics; for example: Coding and Language theory, Automata theory, Combinatorics and Mathematical analysis. The theory of semigroups has been expanding greatly due to its extensive applications in many fields. [6,7,8 and 15] This paper mainly deals with the equivalence relations and congruence relations on semigroups. [5]The relations are useful for understanding the nature of divisibility in a semigroup. [9,10,11,12,13 and 14] In particular Greens relations in semigroups characterizes the elements of a semigroup in terms of the principal ideals they generate. [1]According to J.M. Howie this concept was so all-pervading that a new semigroup can be encountered. Definitions: A semigroup (S, .) is said to be commutative if it satisfies the identity for all in S A semigroup (S, .) is called regular if for each , there exist an element xS such that A semigroup (S, .) is called weakly separative, if x2 = xy = y2 x = y, x,yS. Let S be a semigroup. We define two binary relations E(a), F(a) SS for every aS as 467 | IJREAMV04I09045115 DOI : 10.18231/2454-9150.2018.1223 © 2018, IJREAM All Rights Reserved. International Journal for Research in Engineering Application & Management (IJREAM) ISSN : 2454-9150 Vol-04, Issue-09, Dec 2018 E(a)={(x,y)/ax = ay}, F(a)= {(x,y)/xa = ya}, where x,y S • For a binary relation R SS, we have an element xS, the relation xR = {(xa,xb)/(a,b)R} • Similar way Rx can be defined. • Conider a system (S,), where the relation “” satisfy the following axioms: 1. Reflexivity: a a 2. Symmetry: If a b then b a 3. Transitivity: If a b and b c then a c 4. Anti Symmetric: If a b and b a then a = b. For all a,b,cS If (S,) satisfies (1),(2) and (3) then S is said to possess an Equivalence relation If (S,) satisfies (1),(3) and (4) then S is said to possess a Partial order relation. An equivalence relation with compatible conditions is a congruence relation. Theorem:1 The following relations hold good for a semigroup “S” where a,bS, (i) E(b) E(ab) (ii) F(a) F(ab) (iii) bE(ab) E(a) (iv) F(ab)a F(b) Proof: (i) E(b) E(ab), a,bS By def, E(b) = {(x,y)/ bx = by}, Also E(ab) = {(x,y)/abx = aby}, Now E(b) = {(x,y)/bx = by}, i.e., (x,y)E(b) bx = by abx = aby , aS (x,y)E(ab) Thus E(b) E(ab) (ii) F(a) F(ab) By def, F(a) = {(x,y)/xa = ya} Also F(ab) = {(x,y)/xab = yab} Now F(a) = {(x,y)/xa = ya} i.e., (x,y)F(a) xa = ya xab = yab , For some bS (x,y)F(ab) Thus F(a) F(ab) (iii) bE(ab) E(a) We know E(ab) = {(x,y)/abx = aby}, bE(ab) = {(bx,by)/ abx = aby}, i.e., (bx,by) bE(ab) (x,y) E(ab) 468 | IJREAMV04I09045115 DOI : 10.18231/2454-9150.2018.1223 © 2018, IJREAM All Rights Reserved. International Journal for Research in Engineering Application & Management (IJREAM) ISSN : 2454-9150 Vol-04, Issue-09, Dec 2018 abx = aby a(bx) = a(by) (bx,by) E(a) Thus bE(ab) E(a) (iv) F(ab)a F(b) We know F(ab)= {(x,y)/xab = yab} F(ab)a= {(xa,ya)/xab = yab} i.e., (xa,ya) F(ab)a (x,y)F(ab) xab = yab (xa)b = (ya) (xa,ya) F(b) Thus F(ab)a F(b) Theorem:2 Let us consider another relation SS satisfying the relations E(a) and F(a) such that E(a) = F(a) Proof: Let (x,y) E(a) (x,y) & (x,y) E(a) (x,y) & ax = ay……(1) Let (x,y) F(a) (x,y) & (x,y) F(a) (x,y) & xa = ya……(2) From (1) &(2) (x,y) & (x,y)E(a) (x,y) &(x,y) F(a) i.e., E(a) F(a) Similarly F(a) E(a) Thus E(a) = F(a) Theorem 3: Let (S, .) be regular and E(a), F(a) be two relations on S; then E(a) and F(a) are posets ,if (S,.) is commutative. Proof: (1) We define E(a) on S by (x,y) E(a) ax = ay Put y = x then ax = ax (x,x) E(a) E(a) is reflexive (2) (x,y) E(a) ax = ay and (y,x) E(a) ay = ax Now ax = ay xax = xay ; Also ay = ax yay = yax --- (a) Since S is regular xax = x; Also yay = y --- (b) Now x = xay and y = yax [ From (a) and (b) ] Since S is commutative , xay = yax 469 | IJREAMV04I09045115 DOI : 10.18231/2454-9150.2018.1223 © 2018, IJREAM All Rights Reserved. International Journal for Research in Engineering Application & Management (IJREAM) ISSN : 2454-9150 Vol-04, Issue-09, Dec 2018 Thus we get x = xay = yax = y x = y Therefore, E(a) is anti symmetric (3) Let (x,y) E(a) and (y,z) E(a) i.e., ax = ay and ay = az ax = ay =az ax = az (x,z) E(a) Therefore, E(a) is transitive E(a) is a partial order relation on S Let (x,y) E(a) ax = ay (x, y) E(a) ax = ay ax = ay (S, E(a)) is a poset Theorem 4: E(a) and F(a) are equivalence relations Proof: If (x,y) E(a) then ax = ay , aS Also If (x,y) F(a) then, xa = ya Since ax = ax we have (x,x) E(a) yS i.e., E(a) is reflexive Similarly F(a) is reflexive E(a) and F(a) are reflexive Suppose (x,y) E(a) ax = ay ay = ax (y,x) E(a) i.e., E(a) is symmetric Similarly F(a) is symmetric i.e., E(a) and F(a) are symmetric Now for (x,y) E(a) and (y,z) E(a) ax = ay and ay = az ax = ay = az ax = az E(a) is transitive Similarly, F(a) is also transitive i.e., E(a) and F(a) are transitive Therefore E(a) and F(a) are equivalence relations 470 | IJREAMV04I09045115 DOI : 10.18231/2454-9150.2018.1223 © 2018, IJREAM All Rights Reserved. International Journal for Research in Engineering Application & Management (IJREAM) ISSN : 2454-9150 Vol-04, Issue-09, Dec 2018 Theorem 5: E(a) and F(a) are congruence relations Proof: An equivalence relation with compatible conditions is a congruence relation Now (x,y) E(a) (zx, zy) E(a) for z S and (xz, yz) F(a) i.e.,(x,y) E(a) ax = ay z (ax) = z (ay) (za) x = (za) y (az) x = (az) y a (zx) = a (zy) (zx, zy) E(a) Also (x,y) E(a) ax=ay axz = ayz (xz, yz) E(a) E(a) is a congruence relation Similarly F(a) is a congruence relation. Thus E(a) and F(a) are congruence relations Theorem 6: E(a) is Weakly separative congruence relation: i.e., To Prove, if x2 E(a) xy E(a) y2 x E(a) y Proof: Now x2 E(a) xy a x2 = axy a x x = a x y a x2 y = axy2 ay = ax Also xy E(a) y2 a(xy) = ay2 axyx = a y2 x ax2 y = a y2 x ax=ay Thus E(a) is Weakly separative congruence relation: Theorem 7: E(a) is Least Weakly separative congruence relation on S. Proof: Let „‟ be an arbitrary weakly separative congruence on S such that axym aym+1 Let us take axy2 = ay3 axyy = a y2y a y2x = ay ax = ay 471 | IJREAMV04I09045115 DOI : 10.18231/2454-9150.2018.1223 © 2018, IJREAM All Rights Reserved.
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