Congruences in Hypersemilattices

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1 International Mathematical Forum, Vol. 7, 2012, no. 55, Congruences in Hypersemilattices A. D. Lokhande Yashwantrao Chavan Warna College Warna nagar, Kolhapur, Maharashtra, India Aryani Gangadhara JSPM s Rajarshi Shahu College of Engineering Tathawade, Pune, Maharashtra,India aryani.santosh@gmail.com Abstract In this paper we study congruence relation of hypersemilattices and Homomorphism and isomorphism of hypersemilattices using congruence relation and we prove that hyper meet of two congruence relations is a Congruence relation and is a fixed element of hypersemilattices. Also we prove embedding theorem for hypersemilattices using congruence relation. Finally we prove theorem on family of direct product of hypersemilattices Mathematics Subject Classification: 06B10, 06B99 Keywords: Hypersemilattices, Congruence relation, Homomorphism, Direct Product Introduction The Theory of hyperstructures was introduced in1934 by Marty [1] at the 8 th congress of Scandivinavian Mathematicians. This theory has been subsequently developed by the various authors. Some basic definitions and propositions about the hyperstructures are found in[3].throughout this paper we are using definitions of

2 2736 A. D. Lokhande and Aryani Gangadhara hypersemilattice as discussed in [4].In this paper the concepts of congruence relation are discussed, Also we relate this to isomorphism and homomorphism, that is we prove some results on hypersemilattices using congruence relation. 1. Preliminaries Definition 1.1 [ 4 ]: Let L be a non-empty set and let P(L) denote the Power set of L,P*(L)= P(L)-{ }.A binary operation hyperoperation o on L is a function from L x L to P*(L) and satisfies the following conditions. For all a, b, c Є L and all A,B,C Є P*(L) we have that a o b Є P*(L),C o A = (c o a) Є P*(L), A o C = (a o c ) Є P*(L), A o B = (a o b) Є P*(L). Definition 1.2 [3 ]: Let L be a non-empty set and : L x L P(L) be a hyperoperation,where P(L ) is a power set of L and P*(L) =P(L) -{ } and : L x L L be an operation. Then (L, ) is a hyperlattice if for all a, b,c Є L. 1. a Є a a, a a=a 2. a b=b a, a b=b a 3. (a b) c=a (b c),(a b) c=a ( b c). 4. a Є [a (a b)] [a a b] 5. a Є a b a b=b. Definition 1.3[4] : Hypersemilattices: Let L be a non-empty set with a hyper operation On L satisfying the following conditions, for all a, b, c Є L 1. a Є a a (Idempotent) 2. a b = b a (Commutative) 3. (a b) c = a (b c). (Assosiative) Then (L, ) is called a hypersemilattice. Definition 1.4 [4]: Let (L, ) be a hypersemilattice. An element a Є L is called absorbent element of L if it satisfies c Є a c for all c Є L. An element b Є L is called element of L if it satisfies b c = {b} for all c Є L. Proposition 1.5 [4]: Let (L, ) be a hypersemilattice,then a c a (a c) for all a, c Є L. Definition 1.5 [4]: Let (L, ) and (S, ) be a hypersemilattices. A function f: L is called a homomorphism provided f(a b) = f(a) f(b) for all a, b Є L. If f is S

3 Congruences in hypersemilattices 2737 injective as a map of sets, f is said to be a Monomorphism. If f is surjective, f is called Epimorphism. If f is bijective, f is called an isomorphism. 2. Homomorphism and congruence of Hypersemilattices Theorem 2.1: The homomorphic Image of a hypersemilattice is also a hypersemilattice. Proof: Let f: L S be a homomorphism on L Let a 1, b 1, c 1 f(l),then a, b, c L such that f(a) = a 1, f(b) =b 1, f(c) = c 1. Since L is a hypersemilattice we have a a a f(a) f(a a) f(a) f(a) f(a) a 1 b 1 = f(a) f(b) = f(a b) = f (b a) = f(b) f(a)= b 1 a 1, (a 1 b 1 ) c 1 = [f(a) (b)] f(c) = f[(a b) c] =f [a (b c)] =f(a) [f(b) f(c)] =a 1 (b 1 c 1 ), Hence f(l) is a hypersemi lattice. Definition 2.2 : Let <L, > be a hypersemilattice and θ be an equivalence relation on L. We say that, A θ B if and only if for all a A b B such that a θ b for any A,B L. Definition 2.3: Let L be hypersemilattice, θ is said to be congruence relation if for any a, b, c, d L, a θ b and c θ d imply (a c) θ (b d). We denote the equivalence class {y L/ x θ y} by C x for x L. Theorem 2.4: If θ is a congruence relation on L then, <L/θ, > is a hypersemilattice. Proof: Define the hyperoperation on L/ θ as C x C y = {c t/ t x y}. Since L is a hypersemilattice, x x x and hence C x C x C x, Since x y= y x, we have C x C y = C y C x, Note that (C x C y) C z = { C p /p t z } and C x (C y C z ) = { C q /q x r}. Let C s (C x C y) C z ) then C s { C p /p t z } for some t x y s t z for some t x y s (x y) z C s { C q /q x r } for some r y z C s { C q /q x r }= C x (C y C z ) and hence (C x C y) C z ) C x (C y C z ). Similarly we can prove C x (C y C z ) (C x C y) C z ).Therefore (C x C y) C z ) C x (C y C z ). L/ θ is a hypersemilattice.

4 2738 A. D. Lokhande and Aryani Gangadhara Proposition 2.5: [ C x C y ] Φ/ θ = (C x ) Φ/ θ (C y ) Φ/ θ. Proof: Let C t { [ C x C y ] Φ/ θ / t ( x y ) Φ } t ( x y ) Φ t ( x) Φ ( y ) Φ C t { ( C x ) Φ/ θ ( C y ) Φ/ θ / t ( x) Φ ( y ) Φ } [ C x C y ] Φ/ θ = (C x ) Φ/ θ (C y ) Φ/ θ. Theorem 2.6: Let θ be a congruence relation defined on L then L/ θ is a homomorphic image of the hypersemilattice L. Proof: Let f : L L/ θ be defined by f(x Φ) = [C x ] Φ/ θ.obviously f is well-defined and to show f is homomorphism, consider f(x Φ y Φ) = [C x C y ] Φ/ θ =(C x ) Φ/ θ (C y ) Φ/ θ =f(x Φ) f(y Φ). Therefore f is homomorphism. Select any [C x ] Φ / θ, then x Φ. For this x Φ, f(x Φ) = [C x ] Φ/ θ.f is surjective. Thus there exist an onto homomorphism from L to L/ θ. Hence L/ θ is homomorphic image of L. Definition 2.7: Let f be a homomorphism from a hypersemilattice L 1 to a hypersemilattice L 2.Then Ker f = {x, y L 1 /f(x) =f(y)}. Theorem 2.8: Kernel f is a congruence relation on hypersemilattice L. Proof: First we have to prove kerf is an equivalence relation. As x θ y f(x) = f(y). Obviously kerf is an equivalence relation. We prove Kerf is a congruence relation.let x θ y f(x)=f(y) and a θ b f(a)=f(b). Let us consider f(x a) = f(x) f(a) f(y) f(b) =f(y b). Therefore Ker f is a congruence relation. Theorem 2.9: If h: L 1 L 2 is a surjective homomorphism then there exist an isomorphism f: L 1 /ker h L 2. Proof: Define f: L 1 /ker h L 2 by f (C x )= h(x),where C x is an equivalence class of x under ker h. Clearly f is well-defined. Let C x, C y L 1 /ker h such that f(c x )=h(c y ).Then h(x)= h(y) (x, y) Ker f C x = C y and hence h(x)=h(y) and hence f is one-one. To prove homomorphism, Let f( C x C y ) =f{c t /t x y } ={ h( t) / t x y}= h(x y)=h(x ) h(y) = f(c x ) f(c y ).Hence f is homomorphism. Since h is onto and for any y L 2 there exist C x L 1 /ker h such that f( C x ) = h(x) = y. Hence f is onto. Therefore, L 1 / ker h L 2. Theorem 2.10: product of two congruence relations is a congruence relation.

5 Congruences in hypersemilattices 2739 Proof: Let θ and Φ be any congruence relations defined on Hypersemilattices L and K respectively. Define the relation Ψ on L x K by (C a C b ) Ψ (C c C d ) C a θ C c, C b Φ C d. Then we prove that Ψ is a congruence relation defined on L x K. To prove that (C a C b ) Ψ(C a C b ), as a θ a and C b Φ C b. Therefore (C a C b ) Ψ( Ca C b ). Ψ is reflexive. Let (C a C b ) Ψ(Cc Cd), then C a θ C c and C b Φ C d but θ and Φ and are equivalence relations C c θ C a and C d Φ C b. therefore (C c C d ) Ψ (C a C b ). Ψ is symmetric, Let (C a C b ) Ψ (C c C d) and (C c C d ) and (C x C y )then C a θ C c and C c θ C x, C b Φ C d and C d Φ C y by transitivity of θ and Φ, C a Φ C x and C b Φ C y. ( C a C b ) Ψ (C x C y ).Therefore is transitive. Let (C a1 C b1 ) Ψ ( C c1 C d1 ) and (C a2 C b2 ) Ψ (C c2 C d2 ) This implies C a1 θ C c1 and C b1 Φ C d1 and C a2 θ C c2 and C b2 Φ C d2.hence (C a1 C a2 ) θ (C c1 C c2 ) and (C b1 C b2 ) Φ (C d1 C d2 ).Therefore (C a1 C a2 ) (C b1 C b2 )= (C c1 C d1 ) (Cc 2 Cd 2 ).Therefore Ψ is a congruence relation de fined on L x K. Definition 2.11 : If θ and Φ are congruence relations on L with θ Φ then define a relation Φ/ θ on L/ θ by (C x, C y ) Φ/ θ if and only if (x, y) Φ. Theorem 2.12: Every congruence relation is the kernel of some homomorphism. Proof: Let Φ is a congruence relation on L. Clearly Φ is a Equivalence relation on L/ θ, then to prove Φ/ θ is a congruence relation.let (C x, C y ) and (C z, C w ) Φ/ θ ( x, y),(z, w) Φ, This implies ( x z),(y w) Φ/ θ.that is (C x z, C y w ) Hence Φ/ θ is congruence relation on L/ θ. Now let Φ/ θ is a congruence relation on L/ θ. Similarly we can prove Φ is a congruence relation on L.As Φ is a congruence relation on L this implies (x,y) Φ (C x, C y ) Φ/ θ C x = C y f(c x ) =f(c y ) f(x ) =f(y) Φ is kernel of homomorphism. 3. Hyper Meet of two congruence relations Definition 3.1: Let A be the collection of all congruence relations defined on hyperboolean algebra A. Then hyper meet of two congruence relations is denoted by θ 1 θ 2 and defined as C x (θ 1 θ 2 ) Cy C x θ 1 C y and C x θ 2 C y. Theorem3.2: Let θ 1 and θ 2 be any two congruence relations defined on an hyperboolean algebra A.Define a relation θ 1 θ 2 on A by C x (θ 1 θ 2 ) C y C x θ 1 C y and C x θ 2 C y. Then θ 1 θ 2 is a congruence relation defined on A such that θ 1 θ 2 is the fixed element of θ 1 & θ 2.

6 2740 A. D. Lokhande and Aryani Gangadhara Proof: It is easy to prove that C x (θ 1 θ 2 ) C x as C x θ 1 C x and C x θ 2 C x. C x (θ 1 θ 2 ) C y y (θ 1 θ 2 ) Cy as C x θ 1 C y and C x θ 2 C y. C y θ 1 C x and C y θ 2 C x C x θ 1 C y and C x θ 2 C y. now to prove transitivity, let C x (θ 1 θ 2 ) C y and C y (θ 1 θ 2 ) C z, by definition, C x θ 1 Cy and C x θ 2 C y. C y θ 1 C z and C y θ 2 C z. This implies C x θ 1 C z and C x θ 2 C z. Therefore C x (θ 1 θ 2 ) C z. Let f F, let n be the corresponding integer, Let C xi (θ 1 θ 2 ) C yi for all i This implies C xi θ 1 C yi and C x i θ 2 C yi. That is f (C x 1, C x 2 C x n ) θ 1 f(c y1, C y2,. C yn ) and f(c x 1, C x 2,. C x n ) θ 2 f(c y1, C y2,. C yn ) that is f (C x1, C x2, C. xn) (θ 1 θ 2 ) f(c y1, C y2,. C yn ). θ 1 θ 2 is a hypercongruence relation. To prove that θ 1 θ 2 is fixed element in hypersemilattice L. θ 1 θ 2 θ 1 and θ 1 θ 2 θ 2 is obvious. That is [θ 1 θ 2 ] θ 1 = {θ 1 θ 2 } and [θ 1 θ 2 ] θ 2 = {θ 1 θ 2 }Therefore by [1.4], θ 1 θ 2 is a fixed element of L. Definition3.3 : Con (L)denotes the set of all congruence relations on a hypersemilattice L. Then Con (L) forms complete Lattice with 0 L and 1 L, the fixed (smallest) and (absorbent) Largest lement of congruence relations. Theorem 3.4: For hypersemilattice L with 0 L as fixed element and θ 1, θ 2 Con (L), then there is a natural embedding of L/ θ 1 θ 2 L/ θ 1 x L/ θ 2. Proof: Let Ψ = θ 1 θ 2.Then Φ / θ is a congruence relation on L/ θ 1 θ 2 and let Φ / θ1,φ / θ 2 be congruence relations on L/ θ 1,L/ θ 2 respectively. Define f: L/ θ 1 θ 2 L/ θ 1 x L/ θ 2 by f( (C x ) θ1 θ2 ) ={( (C x ) θ1, (C x ) θ2 ) / x L }.Define a congruence relation Ψ by (C x ), C y ) Φ / θ if and only (x,y) Φ. Let (C x, C y ) and (C z C w ) Φ/ θ ( x, y),(z, w) Φ, This implies ( x z),(y w) Φ/ θ.that is (C x z, C y w ) Hence Φ/ θ is congruence relation on L/ θ. Now let Φ/ θ is a congruence relation on L/ θ. Similarly we can prove Φ/ θ 1 and Φ/ θ 2 are congruence relations on L/ θ 1 and L/ θ 2. f( (C x) θ1 θ2 ) = f( (C y ) θ1 θ2 ) ( (C x) θ1, (C x) θ2 ) =={( (C y ) θ1, (C y ) θ2 ) (C x) θ1 = (C y ) θ1 and (C x) θ2 =(C y ) θ2 (C x, C y ) θ 1 and (C x, C y ) θ 2 (C x, C y ) θ 1 θ 2. But by Theorem [3.2],θ 1 θ 2 is a fixed element. That means by uniqueness property of fixed element of hypersemilattices θ 1 θ 2 = 0 L. Therefore C x = Cy. This implies f is one-one. To prove homomorphism Let f( (C x C y ) θ1 θ2 ) = [ (Cx C y ) θ1, C x C y ) θ2 ] = ( C x, C x ) θ1 θ2 (C y, C y ) θ1 θ2 = ( C x ) θ1 θ2 ( C y ) θ1 θ2.hence f is homomorphism. L/ θ 1 θ 2 L/ θ 1 x L/ θ 2. Corollary 3.5: If hypersemilattice L has congruence relations θ 1 & θ 2 with θ 1 θ 2= 0 L then L L/ θ 1 x L/ θ 2 (an embedding).

7 Congruences in hypersemilattices 2741 Proof: Let Φ be a congruence relation on L and Φ/ θ 1, Φ/ θ 2 be the congruence relations on L/ θ 1 and L/ θ 2 respectively. It can be easily checked using theorem [3.4]. Then define Ψ : L L/ θ 1 x L/ θ 2 by Ψ( x Φ ) = {( (C x) θ1, (C x) θ2 ) / x L } To prove Ψ is one-one let Ψ(x Φ) = Ψ(y Φ) ( (C x) θ1, (C x) θ2 ) =( (C y ) θ1, (C y ) θ2 ).This implies ( (C x) θ1 = (C y ) θ1 ) and ( (C x) θ2 = (C y ) θ2 ), that is (x,y) θ 1 and (x,y) θ 2 therefore (x,y) θ 1 θ 2, but θ 1 θ 2= 0 L implies x=y. Let Ψ (x Φ y Φ ) = [(C x C y ) θ1, (C x C y ) θ2 ]= [(C x) θ1 (C y ) θ1 ] [(C x) θ2 (C y ) θ2 ] = [(C x) θ1, (C x) θ2 ] [(C y ) θ1, (C y ) θ2 ] = Ψ( x Φ ) Ψ( y Φ ). Ψ is a homomorphism. Therefore A A/ θ 1 x A/ θ 2 is an embedding. 4. Direct Product of Hypersemilattices Definition 4.1: Let (L, ) and (S, ) be Hypersemilattices. Define binary operation on the Cartesian product Lx S as follows ( a 1,b 1 ) (a 2,b 2 ) ={(c, d)/c a1 a 2,d b1 b 2 }for all ( a 1,b 1 ), (a 2,b 2 ) L x S,then (L x S, ) is called the direct product of Hypersemilattices (L, ) and (S, ). Definition 4.2: Let {L i / i I} be a family of hypersemilattices. Then the direct product of Li, i I is the Cartesian product (Li/i I) = {(xi), i I/x i L i }. Theorem 4.3: The direct product of family of Hypersemilattices is again a hypersemilattice. Proof: Let {Li/ i I} be a family of hypersemilattices. L = (Li/i I) = {(xi), i I/x i L i }. Define hyperoperation on L as follows: (xi) i I (yi) i I ={(ti) i I/ ti xi yi). It is easy to observe that (xi) i I (xi) i I (xi) i I. (xi) i I (yi) i I =(yi) i I (xi) i I and for any xi, yi, zi L i, i I, ((xi) i I (yi) i I ) (zi) i I = {(pi) i I pi ti zi } and (xi) i I ((yi) i I (zi) i I) = {(ri) i I ri xi qi}.let (si) i I (xi) i I ((yi) i I (zi) i I). Then (si) i I {(pi) i I /pi ti zi } for some ti xi yi si ti zi for some ti xi yi si (xi y) zi si xi (y zi ) si xi qi for some qi yi zi (si) i I {(ri) i I ri xi qi} (si) i I {(ri) i I ri xi qi}=(xi) i I ((yi) i I (zi) i I) and hence ((xi) i I (yi) i I ) (zi) i I (xi) i I ((yi) i I (zi) i I).Similarly we can prove (xi) i I ((yi) i I (zi) i I) ((xi) i I (yi)) i I ) (zi) i I.Hence the theorem.

8 2742 A. D. Lokhande and Aryani Gangadhara References 1. F.Marty, Surune generalization de la notion de group, 8 th Congress Math, (1934)Pages Scandinanes,Stockholm. 2. G.Gratzer General Lattice theory, P.Corsini, Spaces J.Sets P.Sets, Algebraic hyperstructures and applications, Hardonic Press, Inc (1994) P ZHAO Bin, XIAO Ying, HAN Sheng Wei, Hypersemilattices, Received: June, 2012