Microsoft Word - 1. Front cover page with ISSN nos Characterization of Distributive and Standard Ideals in Semilattices Rama Ravi Kumar,E.S.,1* Venkateswara Rao, J2 and Srinivas kumar, V3 1*Department of Mathematics, V. R. Siddhartha Engineering College, Vijayawada, A.P., India (*srrkemani@yahoo.co.in) 2 Department of Mathematics, College of Natural and Computational Sciences, P. O. Box 231, Mekelle University, Mekelle, Ethiopia (venkatjonnalagadda@yahoo.co.in) 3Department of Mathematics, J.N.T.U.H, College of Engineering, J.N.T.U., Kukutpally, Hyderabad - 500 085, A.P., India (srinu_vajha@yahoo.co.in) ABSTRACT This paper investigates the concepts of distributive ideal, dually distributive ideal and standard ideal in a join semilattice. It concerns with the property of ideals in a distributive semilattice. We obtain a characterization theorem for distributive (dually distributive) and standard ideal in a join semilattice. We establish the necessary and sufficient condition for a distributive ideal to be standard ideal. Finally, we bear out the fundamental theorem of homomorphism and Isomorphism theorem of standard ideal. Keywords: Distributive ideal, Distributive semilattice, Dually Distributive ideal, Standard ideal, Join Semi Lattice. [AMS Subject Classification (2000): 06A12, 06A06, A6B99)] 1. INTRODUCTION The concept of distributive ideal, standard ideal and neutral ideal in a lattice L has been introduced and studied by Hashimoto (1952); and Gratzer and Schmidt (1961). Properties of distributive ideals of Birkhoff (1967) are considered in our work. In this paper we studied the notion of distributive (dually) ideal and standard ideal in a semilattice of Gratzer (1978) and produced a characterization theorem of standard ideal. The necessary and sufficient condition for a distributive ideal to be standard ideal was produced. Finally, the fundamental theorem of homomorphism and Isomorphism theorem of standard ideal were proved. 2. METHODOLOGY Hashimoto (1952) and Gratzer and Schmidt (1961) have defined standard and distributive ideal and standard and distributive element in a lattice L and an example of standard ideal as a principal ideal. Also, they afforded a result that if “s” is a standard element and if “a” is an arbitrary element of lattice, then a s is a standard element of the principal ideal (a] and this result is not valid for distributive elements. The properties of distributive ideals Birkhoff (1967) were considered for our work and we investigated the notion of distributive (dually) ideal, © CNCS, Mekelle University ISSN:2220-184X mailto:srrkemani@yahoo.co.in mailto:venkatjonnalagadda@yahoo.co.in http://in.mc83.mail.yahoo.com/mc/compose?to=srinu_vajha@yahoo.co.in Rama Ravi Kumar,E.S., Venkateswara Rao, J and Srinivas Kumar,V (MEJS) Volume 3 (1):20-36, 2011 standard ideal in a semilattice of Gratzer (1978) and produced a characterization theorem of standard ideal. We established the necessary and sufficient condition for a distributive ideal to be standard ideal. Finally, we obtained the fundamental theorem of homomorphism and isomorphism theorem of standard ideal. 3. DISTRIBUTIVE IDEALS 3.1. Definition A semilattice is a partially ordered set (S, ≤) in which any two elements in S have the least upper bound in S. 3.2. Definition A semilattice is a non empty set S with binary operation ∨ defined on it and satisfies the following: Idempotent law : a ∨ a = a for all a in S, Commutative law : a ∨ b = b∨a for all a, b in S, Associative law : a∨ (b∨ c) = (a ∨ b) ∨ c for all a,b,c in S. 3.3. Theorem In a semilattice S, define a ≤ b if and only if a ∨ b = b for all a, b in S. Then (S, ≤) is an ordered set in which every elements has a least upper bound, conversely, given an ordered set P with that property, define a ∨ b = l.u.b.(a, b). Then (P, ≤) is a semilattice. 3.4. Definition A non empty subset D of a semilattice S is called an ideal if (i) for x in D, y in D ⇒ x ∨ y in D , (ii) for x in D, t in S and t ≤ x ⇒ t in D. 3.5. Theorem If I(S) denotes the set of all ideals of a semilattice S, then I(S) is a lattice with respective to the following: (i) D1≤ D2 if and only if D1 ⊆ D2 © CNCS, Mekelle University ISSN:2220-184X 21 Rama Ravi Kumar,E.S., Venkateswara Rao, J and Srinivas Kumar,V (MEJS) Volume 3 (1):20-36, 2011 (ii) D1 ∨ D2 = { x in S / x = x1 ∨ x2 , where x1 is in D1, x2 is in D2} (iii) D1 ∧ D2 = {x in S / x is in D1 and x is in D2}; where D1, D2 are in I(S). 3. 6. Definition The smallest ideal containing x in S is denoted by (x] and is given by (x] = { s in S / s ≤ x}. Such ideal is called principal ideal generated by x. 3.7. Definition An ideal D of a semilattice S is called distributive ideal if and only if D ∨ (X ∧ Y) = (D ∨ X) ∧ (D ∨ Y) for all X, Y in I(S). 3.8. Definition An ideal D of semilattice S is called dually distributive ideal if and only if D ∧ (X ∨ Y) = (D ∧ X) ∨ (D ∧ Y) for all X, Y in I(S). 3.9. Remark The following example shows that an ideal need not be a distributive or dually distributive. Consider the semilattice S = {1,a,b,c,an ……, a1,a0 } given in figure 1. o o c o oo o a0 1 b a Figure 1. Semilattice ideal need not be a distributive (dually distributive). Clearly D={ a0, a1….. an,a}, X={ a0, a1….. an,b}, and Y={ a0, a1….. an,c} are ideals of S. Now X ∧ Y = {a0, a1…….. an} , D ∨ (X ∧ Y) = {a0, a1…….. an, a}, D ∨ X = S and D ∨ Y = S. Therefore (D ∨ X) ∧ ( D ∨ Y) = S and (D ∨ X) ∧ ( D ∨ Y) ≠ D ∨ (X ∧ Y). Hence D is not a distributive ideal of the semilattice S. o a1 an © CNCS, Mekelle University ISSN:2220-184X 22 Rama Ravi Kumar,E.S., Venkateswara Rao, J and Srinivas Kumar,V (MEJS) Volume 3 (1):20-36, 2011 Also X ∨ Y = S , D ∧ (X ∨ Y) = {a0, a1…….. an, a}, D ∧ X = {a0, a1…….. an}, D ∧ Y = {a0, a1…….. an} and (D ∧ X) ∨ (D ∧ Y) = {a0, a1…. an}. Therefore D ∧ (X ∨ Y) ≠ (D ∧ X) ∨ (D ∧ Y) Hence D is not a dually distributive ideal of the semilattice S. 3.10. Result If D1 and D2 are distributive ideals then D1 ∨ D2 is also distributive. 3.10.1. Proof : Let D1 and D2 are distributive ideals of S. Then for any two ideals X and Y of S, (D1 ∨ D2 ) ∨ ( X ∧ Y) = D1 ∨ (D2 ∨ (X ∧ Y)) = D1 ∨ [(D2 ∨ X) ∧ (D2 ∨ Y)] (as D2 is distributive) = [(D1 ∨ (D2 ∨ X)] ∧ [D1 ∨ (D2 ∨ Y)] (as D2 is distributive) = [(D1 ∨ D2) ∨ X] ∧ [(D1 ∨ D2 )∨ Y] Therefore D1 ∨ D2 is a distributive ideal. 3.11. Definition A semilattice S is said to be directed below if a, b ∈ S, then there exists c such that c ≤ a, c ≤ b. 3.12. Definition A semilattice S is called distributive if and only if w ≤ a ∨ b, where w, a, b in S ⇒ there exists x, y in S such that x ≤ a, y ≤ b and w = x ∨ y. 3.13. Theorem A semilattice S is distributive if and only if (i) S is directed below. (ii) The lattice I(S) of all ideals of S is a distributive lattice. 3.13.1. Proof : Suppose a semilattice S is distributive. (i) To prove that S is directed below: © CNCS, Mekelle University ISSN:2220-184X 23 Rama Ravi Kumar,E.S., Venkateswara Rao, J and Srinivas Kumar,V (MEJS) Volume 3 (1):20-36, 2011 Let a, b are in S. Then a ∨ b ∈ S. Since a ≤ a ∨ b and S is distributive there exists x, y in S such that x ≤ a, y ≤ b and a = x ∨ y.Trivially y ≤ x ∨ y = a. Therefore for a, b in S there exists y in S such that y ≤ a, y ≤ b so that S is directed below. (ii) To prove that the lattice I(S) is distributive: Now x ∨ y ∈ D1 ∨ (D2 ∧ D3) ⇔ x ∈ D1, y ∈ (D2 ∧ D3) ⇔ x ∈ D1, y ∈ D2 and y ∈ D3 ⇔ x ∈ D1 , y ∈ D2 and x ∈ D1 , y ∈ D3 ⇔ x ∨ y ∈ D1 ∨ D2 and x ∨ y ∈ D1 ∨ D3 ⇔ x ∨ y ∈ (D1 ∨ D2) ∧ ( D1 ∨ D3). Therefore D1 ∨ ( D2 ∧ D3) = (D1 ∨ D2) ∧ (D1 ∨ D3). Also x ∨ y ∈ (D1 ∧ D2) ∨ (D1 ∧ D3) ⇔ x ∈ D1 ∧ D2 , y ∈ D1 ∧ D3 ⇔ x ∈ D1 and x ∈ D2 , y ∈ D1 and y ∈ D3 ⇔ x ∈ D1 , y ∈ D1 and x ∈ D2 , y ∈ D3 ⇔ x ∨ y ∈ D1 and x ∨ y ∈ D2 ∨ D3 ⇔ x ∨ y ∈ D1 ∧ ( D2 ∨ D3). Therefore D1 ∧ ( D2 ∨ D3) = (D1 ∧ D2) ∨ (D1 ∧ D3) and I(S) is a distributive lattice. Conversely, suppose that S is directed below and I(S) is distributive lattice. Let w ≤ a ∨ b where a, b, w ∈ S. Now (w] = (w] ∧ ((a] ∨ (b])= ((w] ∧ (a]) ∨ ((w] ∧ (b]) = a0 ∨ a1,where a0 ∈ (a], a1 ∈ (b]. Hence there exists a0, a1 in S such that a0 ≤ a; a1 ≤ b and (w] = a0 ∨ a1. Therefore S is distributive semilattice. 3.14. Definition A binary relation θ on a lattice L is called congruence relation if (i) θ is reflexive : x ≡ x (θ) for all x in L (ii) θ is symmetric : x ≡ y (θ) ⇒ y ≡ x (θ) for all x, y in L (iii) θ is transitive : x ≡ y(θ) and y ≡ z(θ) ⇒ x ≡ z(θ) for all x, y, z in L (iv) θ satisfies substitution Property : x ≡ x1(θ) and y ≡ y1(θ) ⇒ x ∨ y ≡ x1 ∨ y1(θ) and x ∧ y ≡ x1 ∧ y1 (θ) for all x, y, x1, y1 in L. 3.15. Theorem Let D be an ideal of semilattice S. Then the following conditions are equivalent. © CNCS, Mekelle University ISSN:2220-184X 24 Rama Ravi Kumar,E.S., Venkateswara Rao, J and Srinivas Kumar,V (MEJS) Volume 3 (1):20-36, 2011 (i) D is distributive. (ii) The map ϕ : X → D ∨ X is a homomorphism of I(S) onto [D) = {X in I(S) / X ≥ D}. (iii) The binary relation θD on I(S) is defined by X ≡ Y (θD) if and only if D ∨ X = D ∨ Y,, where X, Y in I(S) is a congruence relation. 3.15.1. Proof: Let D be an ideal of semilattice S. To prove that (i) ⇒ (ii): Suppose (i) holds. Then D∨ (X ∧Y) = (D ∨ X) ∧ (D ∨ Y) for all X, Y in I(S) Define a map ϕ : X → D ∨ X by ϕ (X) = D ∨ X. → (1) For X, Y in I(S), ϕ ( X ∨ Y) = D ∨ (X ∨ Y) = (D ∨ D) ∨ (X ∨ Y) = D ∨ [D ∨ (X ∨ Y)] = D ∨ [D ∨ X ∨ Y] = D ∨ (D ∨ X) ∨ Y)] = (D ∨ X) ∨ (D ∨ Y) = ϕ (X) ∨ ϕ (Y). Similarly, ϕ (X ∧ Y) = D ∨ (X ∧Y) = (D ∨ X) ∧ (D ∨ Y) = ϕ (X) ∧ ϕ (Y). Therefore ϕ is homomorphism. Next let X in [D). Then X ≥ D so that ϕ (X) = D ∨ X = X. Therefore for any X in [D), there exists X in I(S) such that ϕ (X) = X so that ϕ is homomorphism of I(S) onto [D). To prove (ii) ⇒ (iii): Suppose the map ϕ: X → D ∨ X is a homomorphism of I(S) onto [D) = { X in I(S) / X ≥ D}.Define the binary relation θD in I(S) as X ≡ Y (θD) if and only if D ∨ X = D ∨ Y where X, Y in I(S). We shall show that the relation is congruence: (a) For any X in I(S), D ∨ X = D ∨ X trivially so that X ≡ X (θD) for all X in I(S). Therefore θD is reflexive. (b) For X, Y in I(S), X ≡ Y (θD) ⇒ D ∨ X = D ∨ Y ⇒ D ∨ Y = D ∨ X ⇒ Y ≡ X (θD). Therefore θD is symmetry. (c) For X, Y, Z in I(S), X ≡ Y (θD) and Y ≡ Z (θD) ⇒ D ∨ X = D ∨ Y and D ∨ Y = D ∨ Z ⇒ D ∨ X = D ∨ Z ⇒ X ≡ Z (θD). Therefore θD is Transitive. (d) Substitution Property: Suppose X ≡ X1 (θD) and Y ≡ Y1 (θD) for X, Y, X1, Y1 in I(S). Then D ∨ X = D ∨ X1 and D ∨ Y = D ∨ Y1. → (2) © CNCS, Mekelle University ISSN:2220-184X 25 Rama Ravi Kumar,E.S., Venkateswara Rao, J and Srinivas Kumar,V (MEJS) Volume 3 (1):20-36, 2011 Βy (1) and (2) and since ϕ is a homomorphism, D ∨ (X ∨ Y) = ϕ (X∨ Y) = ϕ (X) ∨ ϕ (Y) , = (D ∨ X ) ∨ (D ∨ Y ) = (D ∨X1 ) ∨ (D ∨ Y1 ) = ϕ (X1) ∨ ϕ (Y1) = ϕ (X1∨ Y1) = D ∨ (X1 ∨ Y1). Therefore X ∨ Y ≡ (X1 ∨ Y1)θD . Similarly we can prove that X ∧ Y ≡ (X1 ∧ Y1)θD. Therefore θD is a congruence relation. To show that (iii) ⇒ (i): Suppose the binary relation θD defined by X ≡ Y (θD) if and only if D ∨ X = D ∨ Y is a congruence relation. For X, Y in I(S), D ∨ (D ∨ X) = (D ∨ D) ∨ X = D ∨ X ⇒ D ∨ X ≡ X (θD) ⇒ X ≡ D ∨ X (θD) by symmetry. Similarly we can prove Y ≡ D ∨ Y (θD). Then by substitution property X ∧ Y ≡ [ ( D ∨ X) ∧ ( D ∨ Y)] (θD). Hence D ∨ ( X ∧ Y) = D ∨ (D ∨ X) ∧ (D ∨ Y) = (D ∨ X) ∧ (D ∨ Y). Therefore D is distributive. 3.16. Result Let D be an ideal of semilattice S. Then by applying the principle of duality to 2.15 we can have the equivalence of the following conditions. (i) D is dually distributive. (ii) The map ϕ : X → D ∧ X is a homomorphism of I (S) onto (D] = {X in I(S) / X ≤ D.} (iii) The binary relation θD on I(S) is defined by X ≡ Y (θD) if and only if D ∧ X = D ∧ Y, where X, Y in I(S) is a congruence relation. 3.17. Definition An ideal D of a semilattice S is called standard ideal if ( ) ( ) (X D Y X D X Y∧ ∨ = ∧ ∨ ∧ ) for all X,Y ∈ I(s). The following example shows that every ideal need not be a standard ideal. © CNCS, Mekelle University ISSN:2220-184X 26 Rama Ravi Kumar,E.S., Venkateswara Rao, J and Srinivas Kumar,V (MEJS) Volume 3 (1):20-36, 2011 3.18. Example Let S = {a0, a1, a2,….. an,a, b,c,d,1} be the semilattice as shown in figure 2 and let D = {a0, a1, a2,….. an,a} S. ⊆ Then for all x, y ∈ D, x ∨ y = a and a ∈D. Next let x ∈ D, t∈S and let t ≤ x. Now t ≤ x and x ∈D implies that t = ai , 0 ≤ i ≤ n or t = a. In either case t a ∈ D and D is an ideal of S. Similarly we can show that X = {a0, a1, a2,….. an, b}and Y = {a0, a1, a2,….. an, c}, are ideals of S. Now X Y∧ = {a0, a1, ….. an} ; = S, D Y∨ X D∧ ={a0, a1,…. an}, ={a(X D Y∧ ∨ ) 0, a1,…. an, b}=X and ( ) ( )X D X Y∧ ∨ ∧ ={a0, a1,…. an}. Therefore ( )X D Y∧ ∨ ≠ ( ) ( )X D X Y∧ ∨ ∧ , shows that X is not a standard ideal. Figure 2. Semilattice ideal need not be a standard ideal. 3.19. Theorem Let L be lattice and let θ be the binary relation on L defined by: x ≡ y(θ) if and only if x ≤ y. If θ is reflexive and symmetric, then θ is a congruence relation if and only if the following three properties are satisfied for all , ,x y z in L. (i) ( ) ( ) ( )x y x y x yθ θ≡ ⇔ ∧ ≡ ∨ (ii) ( ) ( ) ( ), and x y z x y y z x zθ θ θ≤ ≤ ≡ ≡ ⇒ = (iii) ( ) ( ) ( ) and andx y x y x t y tθ θ≡ ≤ ⇒ ∧ = ∧ ( ) ( ) for all t Lx t y t θ∨ = ∨ ∈ © CNCS, Mekelle University ISSN:2220-184X 27 Rama Ravi Kumar,E.S., Venkateswara Rao, J and Srinivas Kumar,V (MEJS) Volume 3 (1):20-36, 2011 3.19.1. Proof: Let the binary relation θ defined on a lattice L by: ( )x y θ≡ if and only if x y≤ be reflexive and symmetric. Assume that θ is a congruence relation. We prove that θ satisfies the properties (i), (ii) and (iii). (i) Let ( )x y θ≡ . Then x y≤ and this implies x y x∧ = and x y y∨ = so that ( ) (x y x y )θ∧ ≡ ∨ .Conversely, suppose ( ) ( )x y x y θ∧ ≡ ∨ . Then x y x y∧ ≤ ∨ and this implies x y x∧ ≤ or ( )x y x θ∧ ≡ . Since θ is symmetric we have ( )x x y θ≡ ∧ . Since θ is a congruence relation, this gives ( )x y θ≡ . (ii) Let x y z≤ ≤ , then x y≤ and y z≤ and this implies ( )x y θ≡ and ( )y z θ≡ . Since θ is a congruence relation θ is transitive, so that ( )x z θ≡ (iii) Let ( )x y θ≡ and x y≤ , then for t ∈ L, x t y t∨ ≤ ∨ implies ( ) ( )x t y t θ∨ ≡ ∨ and similarly x t y t∧ ≤ ∧ for t in L, we have ( )x t y t θ∧ ≡ ∧ . Conversely, suppose θ satisfies the properties (i), (ii) and (iii). We shall show that θ is a congruence relation. Given θ is reflexive and symmetric. Let ( )x y θ≡ and ( )y z θ≡ . Then x y≤ and y z≤ and these imply x y z≤ ≤ . By property (ii) we have ( )x z θ≡ .Therefore θ is transitive. Let ( )1x x θ≡ and ( )1y y θ≡ so that 1 x x≤ and 1y y≤ . This together with the property (iii) gives 1 1x y x y∨ ≡ ∨ and also ( )1 1x y x y θ∧ ≡ ∧ for 1 1, , , Lx y x y ∈ . Thus θ satisfies substitution property. Hence, θ is a congruence relation. 3.20. Theorem Let D be an ideal of a semilattice S. Then, the following conditions are equivalent. (1) D is standard ideal. © CNCS, Mekelle University ISSN:2220-184X 28 Rama Ravi Kumar,E.S., Venkateswara Rao, J and Srinivas Kumar,V (MEJS) Volume 3 (1):20-36, 2011 (2) The binary relation Dθ on I(S) defined by ( )DX Y θ≡ if and only if ( ) 1X Y D X Y∧ ∨ = ∨ for some is a congruence relation. 1D D≤ (3) D is distributive and for all X, Y ∈ I(S) D X D Y, D X =D Y∧ = ∧ ∨ ∨ implies X = Y 3.20.1. Proof: Suppose D is an ideal of a semilattice S. Define the binary relation Dθ on I(S) as ( )DX Y θ≡ if and only if ( ) for some 1X Y D X Y∧ ∨ = ∨ 1D D≤ . (1) => (2) It is sufficient to prove that (i) Dθ is reflexive (ii) Dθ is symmetric (iii) ( )DX Y θ≡ ( ) ( )X Y X Y θ⇔ ∧ ≡ ∨ (iv) ( ) ( )X Y Z, X Y and Y ZD Dθ θ≤ ≤ ≡ ≡ ⇒ ( )X Z Dθ≡ (v) X and Y≤ ( )X Y Dθ≡ ⇒ ( )X Z Y Z Dθ∧ ≡ ∨ for all X, Y, Z ∈ I(s) (i) Let X, Y ∈ I(s) be arbitrary. Then, by the definition of Dθ we have (X ∨ X) ∨ D1 = X ∨ X for 1X=D D≤ ( )X X Dθ⇒ ≡ for all X ∈ I(S). Thus Dθ is reflexive (ii) For X, Y ∈ I(S), X ≡ Y(θD) (X ∧ Y) ∨ D⇒ 1 = X ∨ Y  for some D1 ≤ D. for some D( ) 1Y X D = Y X⇒ ∧ ∨ ∨ 1 ≤ D ( )Y DX θ⇒ ≡ . Thus Dθ is symmetric. (iii) for some D( ) ( ) 1X Y X Y D = X YDθ≡ ⇔ ∧ ∨ ∨ 1 ≤ D ( ) ( ) ( ) ( )1X Y X Y D X Y X Y⇔ ∧ ∧ ∨ ∨ = ∧ ∨ ∨⎡ ⎤⎣ ⎦ for some D1 ≤ D, and by taking we have X=X Y and Y= X Y∧ ∨ ( X ∧ Y ) ∨ D1 = X ∨ Y for some D1 ≤ D ⇔ ( )X Y= X Y Dθ∧ ∨ (iv) Suppose , X Y Z≤ ≤ ( ) ( )X Y and Y ZD Dθ θ≡ ≡ and ( ) 1X Y D X Y⇒ ∧ ∨ = ∨ ( ) 2Y Z D Y Z∧ ∨ = ∨ for D1, D2 ≤ D. Now, 1 2X D = Y and Y D Z∨ ∨ = . © CNCS, Mekelle University ISSN:2220-184X 29 Rama Ravi Kumar,E.S., Venkateswara Rao, J and Srinivas Kumar,V (MEJS) Volume 3 (1):20-36, 2011 Since, X ≤ Y and Y ≤ Z we have, ( ) ( )1 2 1 2X D D X D D∨ ∨ = ∨ ∨ 2 Z∨ = ) = Y D for . 1 2D D D∨ ≤ Then, = Z = X ∨ Z ( ) ( ) (1 2 1 2X Z D D X D D∧ ∨ ∨ = ∨ ∨ Therefore ( )X Z Dθ≡ . (v) Suppose and X Y≤ ( )X Y Dθ≡ ( ) 1X Y D = X Y⇒ ∧ ∨ ∨ for some D1 ≤ D. Since X Y≤ ⇒ X Z Y Z∨ ≤ ∨ ( ) ( )( ) ( ) ( )1X Z Y Z D X Z Y Z⇒ ∨ ∧ ∨ ∨ = ∨ ∨ ∨ Therefore ( ) . ( ) (X Z Y Z Dθ∨ ≡ ∨ ) Similarly we can prove that ( ) ( ) ( )X Z Y Z Dθ∧ ≡ ∧ . Therefore Dθ is a congruence relation. To show that (2) (3): ⇒ Suppose the binary relation Dθ on I(s) defined by ( )X DY θ≡ if and only if for some D( ) 1X Y D = X Y∧ ∨ ∨ 1 ≤ D is a congruence relation. First we prove that D is a Distributive ideal. . For all X,Y∈ I(s) we have X ≤ D X ∨ ⇒ X ∧ ( D X) = X [ X ∧ ( D X)] D = X D (*1) ∨ ⇒ ∨ ∨ ∨ ⇒ Also X (D ∨ X) = X D (*2) ∨ ∨ ⇒ So from (*1) and (*2) we get [X ∧ ( D X)] D = X (D X) ∨ ∨ ∨ ∨ This together with the definition of Dθ implies X ≡ (D X)( ∨ Dθ ) (1) ⇒ Similarly one can show that ( ) ( )Y D Y Dθ≡ ∨ ⇒ (2) Since Dθ is a congruence relation, we have X ∧ Y ≡ [ ( D X) ∧ (D Y)]( ∨ ∨ Dθ )  ( ) ( ) ( ) ( ) ( ) ( )X Y D X D Y D= X Y D X D Y⇒ ∧ ∧ ∨ ∧ ∨ ∨ ∧ ∨ ∨ ∧ ∨⎡ ⎤⎣ ⎦ ( by the def of Dθ ) ( ) ( ) (X Y D= D X D Y⇒ ∧ ∨ ∨ ∧ ∨ ) since ( ) ( )X Y D X D Y∧ ≤ ∨ ∧ ∨ . Therefore ( ) ( ) ( ) ( )X Y D=D X Y D X D Y∧ ∨ ∨ ∨ = ∨ ∧ ∨ for all X, Y ∈ I(S) Therefore D is a distributive ideal. Next let us assume that D X=D Y∧ ∧ and , for all X, Y D X=D Y∨ ∨ ∈ I(S) we shall show that X = Y. © CNCS, Mekelle University ISSN:2220-184X 30 Rama Ravi Kumar,E.S., Venkateswara Rao, J and Srinivas Kumar,V (MEJS) Volume 3 (1):20-36, 2011 From (1) we have ( ) ( ) ( )X Y D X D Y Dθ∧ ≡ ∨ ∧ ∨⎡⎣ ⎤⎦ (since Dθ is congruence relation) ≡ [D ∨ X)  ∧ (D ∨ X)]( Dθ ) (since D ∨ X = D Y) ∨ ( ) (D X )Dθ= ∨ ( )X Dθ≡ (by (1)) But ( )X Y X Dθ∧ ≡ ( )( ) ( )1X Y X D = X Y X⇒ ∧ ∧ ∨ ∧ ∨ for some 1D D≤ ( ) 1X Y D X⇒ ∧ ∨ = . ( since X ∈ I(S) and I(S) is a lattice) (3) ⇒ Also , ( )1 1D X Y D X≤ ∧ ∨ = 1D D≤ ⇒ 1D D X=D Y≤ ∧ ∧ 1D D Y Y⇒ ≤ ∧ ≤ 1D Y⇒ ≤ and . 1 1 1D X, D Y D X Y≤ ≤ ⇒ ≤ ∧ So . But by (3),( ) 1X Y D X Y∧ ∨ = ∧ ( ) 1X Y D X∧ ∨ = . Therefore and im X=X Y∧ plies X ……..(4) Y≤ Similarly, we can show that Y ≤ X ……………(5) Therefore from (4) and (5) we have X = Y. To show that (3) (1): ⇒ Suppose D is distributive and for all X,Y∈ I(s) D X=D Y, D X=D Y∧ ∧ ∨ ∨ implies X = Y. We shall show that D is standard ideal or ( ) ( ) (X D Y X D X Y∧ ∨ = ∧ ∨ ∧ ) for all X,Y∈ I(s). For X, Y ∈ I(S), let B = and C = (X D Y∧ ∨ ) ( ) ( )X D X Y∧ ∨ ∧ . We have (X ∧ D) ( X ∧ Y ) ≤ X and (X ∧ D) ( X ∧ Y ) ≤ D Y so that ∨ ∨ ∨ (X ∧ D) ( X ∧ Y ) ≤ X ∧ ( D Y) or C ≤ B. ∨ ∨ This gives D ∧ C ≤ D ∧ Β …….(1) Now D ∧ X ≤ D and D ∧ X ≤ (D ∧ X) ∨ ( X ∨ Y) = C. ( ) ( ) D X D C D B=D X D Y D D Y X=D X ⇒ ∧ ≤ ∧ ≤ ∧ ∧ ∧ ∨⎡ ⎤⎣ ⎦ = ∧ ∨ ∧ ∧⎡ ⎤⎣ ⎦ Therefore . D B=D C∧ ∧ Also since D is distributive ( ) ( ) ( )( )D B=D (X D Y ) D X D D Y∨ ∨ ∧ ∨ = ∨ ∧ ∨ ∨ ( ) ( )D X D Y= ∨ ∧ ∨ © CNCS, Mekelle University ISSN:2220-184X 31 Rama Ravi Kumar,E.S., Venkateswara Rao, J and Srinivas Kumar,V (MEJS) Volume 3 (1):20-36, 2011 ( )= D X Y∨ ∧ ( )( ) ( )= D D X X Y∨ ∧ ∨ ∧ (by absorption property) ( )( ) ( )= D X D X Y∨ ∧ ∨ ∧ ( ) ( )=D X D X Y D C ∨ ∧ ∨ ∧ = ∨ Therefore . D B = D C∨ ∨ Hence, and . D B=D C∧ ∧ D B = D C∨ ∨ So by (3) B = C and D is a standard ideal. 3.21. Theorem Every standard ideal in a semilattice S is a distributive ideal but converse is not true. 3.21.1. Proof: By the theorem, 2.20 every standard ideal in a semilattice S is a distributive ideal. In the semilattice S = {a0, a1, a2,….. an,a, b,c,d,1}as shown in figure 3 the ideal D = {a0, a1, a2,….. an,1} is a distributive ideal but not a standard ideal. Figure 3. Semilattice distributive ideal is not a standard ideal. 3.22. Theorem The necessary and sufficient condition for a distributive ideal D to be standard in a semilattice S is that D ∧ X = D ∧ Y and D ∨ X = D ∨ Y for all X, Y ∈ I(S) implies X = Y. 3.22.1. Proof: Immediate from the theorem 2.20. © CNCS, Mekelle University ISSN:2220-184X 32 Rama Ravi Kumar,E.S., Venkateswara Rao, J and Srinivas Kumar,V (MEJS) Volume 3 (1):20-36, 2011 3.23 Statement Suppose ϕ is a homomorphism of a semilattice S on to a semilattice S1 and D is a standard ideal of S. The binary relation θD defined by x ≡ y(θD) if and only if ϕ(x) = ϕ(y) where x, y ∈ S, is such that (i) θD is a congruence relation on S (ii) S/θD is a semilattice (iii) S/θD ≅ S1 3.23.1. Proof: (i) First let us show that θD is a congruence relation on S. Since ϕ(x) = ϕ(x) for x ∈ S, by definition of θD, we have x ≡ x (θD). Thus θD is a reflexive. Suppose x ≡ y(θD) for x, y ∈ S.Then ϕ(x) = ϕ(y) or ϕ(y) = ϕ(x), which implies that y ≡ x (θD). Thus θD is symmetric. Suppose x ≡ y(θD) and y ≡ z(θD) for x, y, z ∈ S. Then ϕ (x) = ϕ(y) and ϕ(y) = ϕ(z) so that ϕ(x) = ϕ(z) which implies x ≡ z(θD). Thus θD is transitive. Suppose x ≡ x1 (θ) and y ≡ y1(θ) .Then we have ϕ (x) = ϕ(x1) and ϕ(y) = ϕ(y1) Now ϕ (x ∨ y) = ϕ(x) ∨ ϕ(y) (as ϕ is homomorphism) = ϕ(x1) ∨ ϕ(y1) = ϕ(x1 ∨ y1) (as ϕ is homomorphism). This implies x ∨ y ≡ (x1 ∨ y1) (θD). Similarly, ϕ(x ∧ y) = ϕ(x) ∧ ϕ(y) = ϕ(x1) ∧ ϕ(y1)=ϕ(x1 ∧ y1) implies x ∧ y ≡ x1 ∧ y1(θD). Therefore θD satisfies substitution property. Hence θD is a congruence relation. (ii) To prove S/θD is a semilattice let S/θD = { [x] θD / x ∈ S}. Define ∨ on S/θD by [x] (θD) ∨ [y] (θD) = (x ∨ y) (θD) where [x] (θD), [y] (θD) ∈ S/θD. Since x, y ∈ S, x ∨ y ∈ S as S is a semilattice which implies (x ∨ y) (θD) ∈ S/θD. Therefore S/θD is a semilattice. (iii) To prove S/θD ≅ S1, let us define a map ψ : S/θD → S1 by ψ ([x] (θD)) = ϕ(x) for [x] (θD) ∈ S/θD. © CNCS, Mekelle University ISSN:2220-184X 33 Rama Ravi Kumar,E.S., Venkateswara Rao, J and Srinivas Kumar,V (MEJS) Volume 3 (1):20-36, 2011 For [x] (θD), [y] (θD) ∈ S/θD, [x] (θD) = [y] (θD) ⇒ x ≡ y(θD) ϕ(x) = ϕ(y) ⇒ ⇒ ψ ([x] (θD)) = ψ ([y] (θD)). Therefore ψ is well defined. Further, for [x] (θD), [y] (θD) ∈ S/θD, ψ ([x] (θD)) = ψ ([y] (θD)) ⇒ ϕ(x) = ϕ(y) ⇒ x ≡ y (θD) [x] (θ⇒ D) = [y] (θD). This show ψ is one-one. Let z1∈ S1. Then there exists z ∈ S such that ϕ(z) = z1, since ϕ is onto. So [z] (θD) ∈ S/θD and ψ ([z] (θD)) = φ(z) =z1. Therefore, for z1∈ S1, there exists [z] (θD) ∈ S/θD ,such that ψ ([z] (θD)) = z1 so that ψ is onto. Finally, let us show that ψ is homomorphism. For [x] (θD), [y] (θD) ∈ S/θD we have ψ ([x] (θD)) ∨ [y] (θD)) = ψ ((x ∨ y) (θD)) = ϕ (x ∨ y) = ϕ(x) ∨ ϕ(y) = ψ ([x] (θD)) ∨ ψ ([y] (θD)). Therefore ψ is an onto homomorphism.Hence S/θD ≅ S1. 3.24. Theorem Let S be a semilattice, I is an ideal of S and D is a standard ideal of S such that D ⊆ I. Then (i) I is a standard ideal in S and if and only if I/D is a standard ideal in S/D (ii) S/I ≅ (S/D) / (I/D) 3.24.1. Proof: Let S be a semilattice, I is an ideal of S and D a standard ideal of S such that D ⊆ I. (i) Let I be a standard ideal in S. To prove that I/D is a standard ideal in S/D, it is sufficient to prove that I/D is the homomorphic image of I. Now, define ϕ : S → S/D by ϕ(x) = [x] θD, where x ∈ S. As in theorem 2.23 one can see that ϕ is an onto homomorphism. If we restrict ϕ from I to I/D, we have ϕ(I) is an onto homomorphic image of I and ϕ(I) = I/D, which implies ϕ(I) = I/D is a standard ideal. Conversely suppose that I/D is a standard ideal of S/D. © CNCS, Mekelle University ISSN:2220-184X 34 Rama Ravi Kumar,E.S., Venkateswara Rao, J and Srinivas Kumar,V (MEJS) Volume 3 (1):20-36, 2011 For X, Y ∈ I(S), let Y ,X be the homomorphic images of X and Y respectively under the map ϕ : S → S/D. Since I/D is a standard ideal in S/D, we have YX ≡ (θI/D) (from characterization theorem for standard ideal) ⇒ ( ) YXIY X 1 ∨=∨∧ for some I/DII1 =≤ ⇒ (X ∧ Y) ∨ I1 = X ∨ Y for some I1 ≤ I. ⇒ X ≡ Y(θI) ⇒ I is a standard ideal in S. (ii) To prove that S/I ≅ (S/D)/(I/D) define g: S → (S/D)/(I/D) by g(x) = [ x ] θ(I/D) where x ∈ S. For x = y where x, y ∈ S. ⇒ [ x ] θ(I/D) = [ y ] θ(I/D) ⇒ g(x) = g(y). Therefore g is well defined. To show that g is onto, let [ x ] θ(I/D) ∈ (S/D)/(I/D). Then x ∈ S/D for some x ∈ S and g(x) = [ x ]θ(I/D) . Therefore g is onto. Finally, for x, y ∈ S, g(x ∨ y) = [ YX ∨ ] θ(I/D) = [ X ] θ(I/D) ∨ [ Y ] θ(I/D) = g(x) ∨ g(y). This shows that g is a homomorphism Clearly ker g = I, so that by fundamental theorem of homomorphism S/I ≅ (S/D)/(I/D). 3.25. Theorem A semilattice S is distributive ⇔ Every ideal D of S is a standard ideal. 3.25.1. Proof: Assume that in a semilattice S every ideal D is a standard ideal. Then by the theorem 2.20, D is a distributive ideal and ( ) ( ) ( )D X Y D X D Y∨ ∧ = ∨ ∧ ∨ for all X, Y ∈ I(s). This is true for all D so that I(s) is a distributive lattice. This implies that S is a distributive semilattce by Theorem 2.13 Conversely, suppose that a semilattice S is a distributive semillatice and D is an ideal of S. Now S is a distributive semilattice of I(s) ⇒ I(s) is a distributive lattice, by Theorem 2.13 ⇒ Every element in I(s) is standard, since I(s) does not contain N5 or M3 ⇒ Every ideal D of S is a standard ideal. 4. CONCLUSION © CNCS, Mekelle University ISSN:2220-184X 35 Rama Ravi Kumar,E.S., Venkateswara Rao, J and Srinivas Kumar,V (MEJS) Volume 3 (1):20-36, 2011 In this paper, we investigated the notions of distributive (dually) ideal and standard ideal in a semilattice, and established a characterization theorem of standard ideal. We ascertain that set of all ideals of a semilattice is a lattice. We attain the equivalent conditions for a semilattice (ideal of a semilattice) to be distributive (dually distributive). We confirm that every ideal need not be a standard ideal. We define a congruence relation on a lattice and achieve its equivalent conditions. We get hold of the equivalent conditions for an ideal of a semilattice to be a standard ideal. We set up that every standard ideal in a semilattice is a distributive ideal but converse is not true. We take the necessary and sufficient condition for a distributive ideal to be standard in a semilattice. We concluded with the result that a semilattice is distributive if and only if every ideal of it is a standard ideal. 5. REFERENCES Birkhoff, G. 1967. Lattice theory. Amer. Math. Soc., Callog Publication, XXV Providence, R.I. Gratzer, G.1962. A characterization of Neutral elements in Lattices. Magyar Tub Akab. Mat. Kutato Int Kozi., 7:191-192. Gratzer, G. 1978. General Lattice theory. Academic press Inc. Gratzer, G & Schmidt, E.T.1961.Standard ideals in lattices. Acta Math. Sci. Hung., 12: 17-86. Hashimoto, J. 1952. Ideal theory for Lattices. Math. Japan, 2:149-186. Hossain, M.A & Noor, A. S. A. 2007. Modular and Standard filters of a Directed above Meet Semilattice. Thammasat. Int.J.Sc.Tech., 12(4):1-5. © CNCS, Mekelle University ISSN:2220-184X 36