International Mathematical Forum, Vol. 7, 2012, no. 55, 2735 - 2742

Congruences in Hypersemilattices

A. D. Lokhande

Yashwantrao Chavan Warna College

Warna nagar, Kolhapur, Maharashtra, India e-mail:aroonlokhande@gmail.com

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 8th

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

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 S

is called a homomorphism provided f(a ⊗ b) = f(a) ⊗ f(b) for all a, b Є L. If f is

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 a1, b1, c1 ∈ f(L),Then ∋ a, b, c ∈ L such that f(a) = a1, f(b) =b1, f(c) = c1. Since

L is a hypersemilattice we have ⇒ a ∈ a ⊗ a ⇒ f(a) ∈ f(a ⊗ a) f(a) ∈ f(a) ⊗

f(a) a1 ⊗ b1 = f(a) ⊗ f(b) = f(a ⊗ b) = f (b ⊗ a) = f(b) ⊗ f(a)= b1 ⊗ a1 , (a1 ⊗

b1) ⊗ c1 = [f(a) ⊗ (b)] ⊗ f(c) = f[(a ⊗ b) ⊗ c] =f [a ⊗ (b ⊗ c)] =f(a) ⊗

[f(b) ⊗ f(c)] =a1 ⊗ (b1 ⊗ c1), 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 Cx 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 Cx ⊗ Cy = {ct/ t ∈ x⊗y}. Since L is a

hypersemilattice, x ∈ x ⊗ x and hence Cx ∈ Cx ⊗ Cx, Since x ⊗ y= y ⊗ x, we have

Cx ⊗Cy = Cy ⊗ Cx, Note that (Cx ⊗ Cy) ⊗ Cz = { Cp/p ∈ t ⊗ z } and Cx ⊗

(Cy ⊗ Cz ) = { Cq/q ∈x ⊗ r}. Let Cs ∈ (Cx ⊗Cy) ⊗ Cz) then Cs ∈ { Cp/p

∈ t ⊗ z } for some t ∈ x ⊗ y s ∈ t ⊗ z for some t ∈ x ⊗ y s ∈ (x ⊗ y) ⊗ z

Cs ∈ { Cq/q ∈x ⊗ r } for some r ∈ y⊗ z Cs ∈ { Cq/q ∈x ⊗ r }= Cx ⊗

(Cy ⊗ Cz ) and hence (Cx ⊗ Cy) ⊗ Cz ) Cx ⊗ (Cy ⊗ Cz ). Similarly we can

prove Cx ⊗ (Cy ⊗ Cz ) (Cx ⊗ Cy) ⊗ Cz ) .Therefore (Cx ⊗ Cy) ⊗ Cz ) Cx ⊗

(Cy ⊗ Cz ). L/ θ is a hypersemilattice.

2738 A. D. Lokhande and Aryani Gangadhara

Proposition 2.5: [ C x ⊗ Cy] Φ/ θ = (Cx) Φ/ θ ⊗ (Cy) Φ/ θ.

Proof: Let Ct ∈ { [ Cx ⊗ Cy] Φ/ θ / t ∈ ( x ⊗ y ) Φ } ⇔ t ∈ ( x ⊗ y ) Φ ⇔ t ∈ ( x) Φ ⊗

( y ) Φ ⇔ Ct ∈ { ( Cx ) Φ/ θ ⊗ ( Cy) Φ/ θ / t ∈ ( x) Φ ⊗ ( y ) Φ }⇔ [ Cx ⊗ Cy] Φ/ θ = (Cx)

Φ/ θ ⊗ (Cy) 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) = [Cx] Φ/ θ .Obviously f is well-defined

and to show f is homomorphism, consider f(x Φ ⊗ y Φ) = [Cx ⊗ Cy] Φ/ θ =(Cx) Φ/ θ ⊗

(Cy) Φ/ θ =f(x Φ) ⊗ f(y Φ). Therefore f is homomorphism. Select any [Cx] ∈ Φ / θ,

then x ∈ Φ. For this x ∈ Φ , f(x Φ) = [Cx] 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 L1 to a

hypersemilattice L2.Then Ker f = {x, y ∈ L1/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: L1 L2 is a surjective homomorphism then there exist an

isomorphism f: L1/ker h L2.

Proof: Define f: L1/ker h L2 by f (Cx)= h(x),where Cx is an equivalence class of x

under ker h. Clearly f is well-defined. Let Cx, Cy ∈ L1/ker h such that

f(Cx)=h(Cy).Then h(x)= h(y) (x, y) ∈ Ker f

Cx= Cy and hence h(x)=h(y) and

hence f is one-one. To prove homomorphism, Let f( Cx ⊗ Cy) =f{Ct /t ∈ x ⊗ y } ={ h(

t) / t ∈ x ⊗ y}= h(x ⊗ y)=h(x ) ⊗ h(y) = f(Cx) ⊗ f(Cy).Hence f is homomorphism.

Since h is onto and for any y ∈ L2 there exist Cx ∈ L1/ker h such that f( Cx) = h(x) =

y. Hence f is onto. Therefore, L1/ ker h L2.

Theorem 2.10: product of two congruence relations is a congruence relation.

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 (Ca ⊗ Cb) Ψ (Cc ⊗ Cd) Ca θ Cc ,

Cb Φ Cd. Then we prove that Ψ is a congruence relation defined on L x K . To prove

that (Ca ⊗ Cb) Ψ(Ca ⊗ Cb), as a θ a and Cb Φ Cb. Therefore (Ca ⊗ Cb) Ψ(Ca ⊗ Cb). Ψ

is reflexive. Let (Ca ⊗ Cb) Ψ(Cc ⊗ Cd), then Ca θ Cc and Cb Φ Cd but θ and Φ and are

equivalence relations Cc θ Ca and Cd Φ Cb. therefore (Cc ⊗ Cd) Ψ (Ca⊗ Cb). Ψ is

symmetric, Let (Ca ⊗ Cb) Ψ (Cc ⊗ C d) and (Cc ⊗ Cd) and (Cx ⊗ Cy )then Ca θ Cc and

Cc θ Cx, Cb Φ Cd and Cd Φ Cy by transitivity of θ and Φ , Ca Φ Cx and Cb Φ Cy. ( Ca ⊗

Cb) Ψ (Cx⊗ Cy).Therefore is transitive. Let (Ca1 ⊗ Cb1) Ψ ( Cc1 ⊗ Cd1) and (Ca2 ⊗ Cb2)

Ψ (Cc2 ⊗ Cd2) This implies Ca1 θ Cc1 and Cb1 Φ Cd1 and Ca2 θ Cc2 and Cb2 Φ

Cd2.Hence (Ca1⊗ Ca2) θ (Cc1 ⊗ Cc2) and (Cb1 ⊗ Cb2) Φ (Cd1 ⊗ Cd2).Therefore (Ca1 ⊗Ca2)

⊗ (Cb1⊗Cb2)= (Cc1⊗Cd1) ⊗(Cc2⊗ Cd2) .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 (Cx , Cy ) ∈ 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 (Cx , Cy ) and (Cz , Cw) ∈ Φ/ θ

( x, y) ,(z, w) ∈ Φ, This implies ( x ⊗ z) ,(y ⊗ w) ∈ Φ/ θ .That is (Cx ⊗z , Cy⊗ 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 ⇔ (Cx , Cy) ∈ Φ/ θ ⇔ Cx = Cy ⇔ f(Cx) =f(Cy)

⇔ 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

θ1 ⊗ θ2 and defined as Cx (θ1 ⊗ θ2) Cy

C x θ1 Cy and C x θ2 Cy.

Theorem3.2: Let θ1 and θ2 be any two congruence relations defined on an

hyperboolean algebra A .Define a relation θ1 ⊗ θ2 on A by C x (θ1 ⊗ θ2) C y ⇔ C x

θ1C y and C x θ2 Cy. Then θ1 ⊗ θ2 is a congruence relation defined on A such that θ1

⊗ θ2 is the fixed element of θ1 & θ2.

2740 A. D. Lokhande and Aryani Gangadhara

Proof: It is easy to prove that C x (θ1 ⊗ θ2) C x as C

x θ1 C x and C x θ2 C x. C x (θ1 ⊗

θ2) Cy y (θ1 ⊗ θ2) Cy as

C x θ1 Cy and C x θ2 Cy.

Cyθ1 C x and Cyθ2 C x

C x

θ1 Cy and C x θ2 Cy. now to prove transitivity, let C x (θ1 ⊗ θ2) Cy and Cy (θ1 ⊗ θ2) Cz,

by definition , C x θ1 Cy and C x θ2C y. Cy θ1 Cz and Cy θ2 Cz. This implies C x θ1 Cz and

C x θ2C z. Therefore C x (θ1 ⊗ θ2) Cz. Let f ∈F, let n be the corresponding integer, Let C xi (θ1 ⊗ θ2) Cyi for all i This implies C xi θ1 Cyi and C x i θ2 Cyi. That is f (C x 1, C x

2… C x n ) θ1f(Cy1, Cy2,…. Cyn ) and f(C x 1, C x 2,…. C x n ) θ2f(Cy1, Cy2,…. Cyn) that is

f (C x1, C x2,… C. xn) (θ1 ⊗ θ2) f(Cy1, Cy2,…. Cyn) . θ1 ⊗ θ2 is a hypercongruence

relation. To prove that θ1 ⊗ θ2 is fixed element in hypersemilattice L. θ1 ⊗ θ2 θ1and

θ1 ⊗ θ2 θ2 is obvious. That is [θ1 ⊗ θ2] ⊗ θ1 = {θ1 ⊗ θ2 } and [θ1 ⊗ θ2] ⊗ θ2 = {θ1 ⊗

θ2 }Therefore by [1.4], θ1 ⊗ θ2 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 0L and 1L, the fixed

(smallest) and (absorbent) Largest lement of congruence relations.

Theorem 3.4: For hypersemilattice L with 0L as fixed element and θ1, θ2 ∈ Con (L), then there is a natural embedding of L/ θ1 ⊗ θ2 L/ θ1 x L/ θ2.

Proof: Let Ψ = θ1 ⊗ θ2 .Then Φ / θ is a congruence relation on L/ θ1 ⊗ θ2 and let Φ

/ θ1,Φ / θ2 be congruence relations on L/ θ1 ,L/ θ2 respectively. Define f: L/ θ1 ⊗ θ2

L/ θ1 x L/ θ2 by f( (Cx) θ1 ⊗

θ2 ) ={( (Cx)

θ1 , (Cx)

θ2 ) / x ∈ L } .Define a

congruence relation Ψ by (Cx), Cy) ∈ Φ / θ if and only (x ,y) ∈ Φ. Let (Cx , Cy )

and (Cz Cw) ∈ Φ/ θ ( x, y) ,(z, w) ∈ Φ, This implies ( x ⊗ z) ,(y ⊗ w) ∈ Φ/ θ

.That is (Cx⊗z , Cy⊗ 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/ θ1 and Φ/ θ2 are congruence

relations on L/ θ1 and L/ θ2. f( (C x) θ1 ⊗

θ2 ) = f( (Cy)

θ1 ⊗

θ2 )

( (C x)

θ1 , (C x)

θ2 )

=={( (Cy) θ1 , (Cy)

θ2 ) (C x)

θ1 = (Cy)

θ1 and (C x)

θ2 =(Cy)

θ2 (C x , Cy)

∈ θ1 and

(C x, Cy) ∈ θ2 (C x , Cy)

∈ θ1 ⊗ θ2 .But by Theorem [3.2],θ1 ⊗ θ2 is a fixed

element. That means by uniqueness property of fixed element of hypersemilattices θ1 ⊗ θ2 = 0L. Therefore C x = Cy. This implies f is one-one. To prove homomorphism

Let f( (C x ⊗ Cy ) θ1 ⊗

θ2) = [ (Cx ⊗ Cy )

θ1 , C x ⊗ Cy )

θ2 ] = ( C x , C x )

θ1 ⊗ θ2 ⊗ (Cy ,

Cy ) θ1 ⊗ θ2 = ( C x )

θ1 ⊗ θ2 ⊗ ( Cy )

θ1 ⊗ θ2.Hence f is homomorphism. L/ θ1 ⊗ θ2 L/ θ1

x L/ θ2.

Corollary 3.5: If hypersemilattice L has congruence relations θ1 & θ2 with θ1 ⊗ θ2=

0L then L L/ θ1 x L/ θ2 (an embedding).

Congruences in hypersemilattices 2741

Proof: Let Φ be a congruence relation on L and Φ/ θ1 , Φ/ θ2 be the congruence

relations on L/ θ1 and L/ θ2 respectively. It can be easily checked using theorem [3.4].

Then define Ψ : L L/ θ1 x L/ θ2 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 ) =( (Cy)

θ1 , (Cy)

θ2 )

.This

implies ( (C x) θ1 = (Cy)

θ1 ) and

( (C x)

θ2 = (Cy)

θ2 ) , that is (x ,y) ∈ θ1 and (x ,y) ∈ θ2

therefore (x ,y) ∈ θ1 ⊗ θ2, but θ1 ⊗ θ2= 0L implies x=y. Let Ψ (x Φ ⊗ y Φ) = [(C x ⊗

Cy) θ1 , (C x ⊗ Cy)

θ2 ]= [(C x)

θ1 ⊗ (Cy)

θ1 ] ⊗ [(C x)

θ2 ⊗ (Cy)

θ2] = [(C x)

θ1 , (C x)

θ2

] ⊗ [(Cy) θ1 , (Cy)

θ2] = Ψ( x Φ) ⊗ Ψ( yΦ). Ψ is a homomorphism. Therefore A A/

θ1 x A/ θ2 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 ( a1,b1) •(a2,b2) ={(c, d)/c ∈ a1 ⊗ a2,d ∈b1⊗

b2}for all ( a1,b1), (a2,b2) ∈ L x S ,then (L x S, •) is called the direct product of Hypersemilattices (L, ⊗ ) and (S, ⊗ ).

Definition 4.2: Let {Li/ 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/xi ∈ Li}.

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/xi

∈ Li}. 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 ∈ Li , 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.

2742 A. D. Lokhande and Aryani Gangadhara

References

1. F.Marty, “Surune generalization de la notion de group” , 8

th Congress Math,

(1934)Pages 45-49 Scandinanes,Stockholm.

2. G.Gratzer “General Lattice theory”, 1998.

3. P.Corsini, Spaces J.Sets P.Sets, “Algebraic hyperstructures and applications”,

Hardonic Press, Inc (1994) P-45-53.

4. ZHAO Bin, XIAO Ying, HAN Sheng Wei,“Hypersemilattices”,

http://www.paper.edu.cn

Received: June, 2012