A Study on (Q,L)-Fuzzy Normal Subsemiring of a Semiring
S. Sampathu^{1}, S. Anita Shanthi^{2}, A. Praveen Prakash^{3}
^{1}Department of Mathematics, Sri Muthukumaran College of Education, Chikkarayapuram, Chennai, Tamil Nadu, India
^{2}Department of Mathematics, Annamalai University, Tamil Nadu, India
^{3}Department of Mathematics, Hindustan University, Padur, Tamil Nadu, India
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To cite this article:
S. Sampathu, S. Anita Shanthi, A. Praveen Prakash. A Study on (Q,L)-Fuzzy Normal Subsemiring of a Semiring. American Journal of Applied Mathematics. Vol. 3, No. 4, 2015, pp. 185-188. doi: 10.11648/j.ajam.20150304.14
Abstract: In this paper, we introduce the concept of (Q,L)-fuzzy normal subsemirings of a semiring and establish some results on these. We also made an attempt to study the properties of (Q,L)-fuzzy normal subsemirings of semiring under homomorphism and anti-homomorphism , and study the main theorem for this. We shall also give new results on this subject.
Keywords: (Q,L)-Fuzzy Subset, (Q,L)-Fuzzy Subsemiring, (Q,L)-Fuzzy Normal Subsemiring, Product Of (Q,L)-Fuzzy Subsets, Strongest (Q, L)-Fuzzy Relation, Pseudo (Q, L)-Fuzzy Coset
1. Introduction
There are many concepts of universal algebras generalizing an associative ring ( R ;+; . ). Some of them in particular, nearrings and several kinds of semirings have been proven very useful. An algebra (R ; +, .) is said to be a semiring if (R;+) and (R; .) are semigroups satisfying a.(b+c)=a.b+a.c and (b+c).a=b.a+c.a for all a, b and c in R. A semiring R is said to be additively commutative if a+b = b+a for all a, b in R. A semiring R may have an identity 1, defined by 1. a = a = a. 1 and a zero 0, defined by 0+a=a=a+0 and a.0=0=0.a for all a in R. After the introduction of fuzzy sets by L.A.Zadeh[15], several researchers explored on the generalization of the concept of fuzzy sets. The notion of fuzzy subnearrings and ideals was introduced by S.Abou Zaid[10]. A.Solairaju and R.Nagarajan [12] have introduced and defined a new algebraic structure called Q-fuzzy subgroups. In this paper, we introduce the concept of (Q,L)-fuzzy normal subsemiring of a semiring and established some results.
2. Preliminaries
2.1. Definition 1
Let X be a non-empty set and L = (L, ≤) be a lattice with least element 0 and greatest element 1 and Q be a non-empty set. A (Q, L)-fuzzy subset A of X is a function A: X×Q → L.
2.2. Definition 2
Let ( R, +, ∙ ) be a semiring and Q be a non empty set. A (Q, L)-fuzzy subset A of R is said to be a (Q, L)-fuzzy subsemiring (QLFSSR) of R if the following conditions are satisfied:
(i) A( x+y, q ) ≥ A(x, q) ˄ A(y, q),
(ii) A( xy, q ) ≥ A(x, q) ˄ A(y, q), for all x and y in R and q in Q.
2.3. Definition 3
Let R be a semiring and Q be a non-empty set. An (Q, L)-fuzzy subsemiring A of R is said to be an (Q, L)-fuzzy normal subsemiring (QLFNSSR) of R if it satisfies the following conditions:
(i) A(x+y,q) = A(y+x,q),
(ii) A(xy,q) = A(yx,q), for all x and y in R and q in Q.
2.4. Definition 4
Let A and B be any two (Q,L)-fuzzy subsets of sets G and H, respectively. The product of A and B, denoted by A×B, is defined as A×B={<(( x, y), q),A×B((x,y),q)>/ for all x in R and y in H and q in Q}, where A×B((x,y),q)=A(x, q)˄ B(y, q).
2.5. Definition 5
Let (R,+,∙ ) and (R^{׀},+,∙ ) be any two semirings and Q be a non empty set. Let f:R→R^{׀} be any function and A be a (Q,L)-fuzzy subsemiring in R, V be a (Q,L)-fuzzy subsemiring in f(R)=R^{׀}, defined by V(y,q) =A(x,q), for all x in R and y in R^{׀ }and q in Q. Then A is called a pre-image of V under f and is denoted by f ^{-1}(V).
2.6. Definition 6
Let A be a (Q,L)-fuzzy subset in a set S, the strongest (Q, L)-fuzzy relation on S, that is a (Q,L)-fuzzy relation V with respect to A given by V((x,y),q) = A(x,q)˄A(y,q), for all x and y in S and q in Q.
2.7. Definition 7
A (Q,L)-fuzzy subset A of a set X is said to be normalized if there exists an element x in X such that A(x,q)=1.
2.8. Definition 8
Let A be an (Q, L)-fuzzy subsemiring of a semiring (R, +, ∙ ) and a in R. Then the pseudo (Q, L)-fuzzy coset (aA)^{p} is defined by ( (aA)^{p})(x,q) = p(a)A(x,q), for every x in R and for some p in P and q in Q.
2.9. Definition 9
Let A be a (Q,L)-fuzzy subset of X. For a in L, a Q-level subset of A is the set A_{a} = { xÎX : A(x,q) ≥ a}.
3. Properties of (Q,L)-Fuzzy Normal Subsemiring of a Semiring
3.1. Theorem 1
Let (R,+,.) be a semiring and Q be a non-empty set. If A and B are two (Q,L)-fuzzy normal subsemirings of R, then their intersection A∩B is an (Q,L)-fuzzy normal subsemiring of R.
Proof: Let x and yÎR. Let A={á(x,q),A(x,q)ñ/ x in R and q in Q} and B={á(x,q), B(x,q)ñ/ x in R and q in Q} be (Q,L)-fuzzy normal subsemirings of a semiring R. Let C=A∩B and C={á(x,q),C(x,q)ñ/x in R and q in Q}. Then, Clearly C is an (Q,L)-fuzzy subsemiring of a semiring R, since A and B are two (Q,L)-fuzzy subsemirings of a semiring R. And (i) C(x+y,q)=A(x+y,q)˄B(x+y,q)=A(y+x,q)˄B(y+x,q) =C(y+x,q), for all x and y in R and q in Q. Therefore,C(x+y,q) =(y+x,q), for all x and y in R and q in Q. (ii) C(xy,q) = A(xy,q)˄B(xy,q)=A(yx,q)˄B(yx,q)=C(yx,q), for all x and y in R and q in Q. Therefore, C(xy,q)=C(yx,q), for all x and y in R and q in Q. Hence A∩B is an (Q,L)-fuzzy normal subsemiring of a semiring R.
3.2. Theorem 2
Let R be a semiring and Q be a non-empty set. The intersection of a family of (Q,L)-fuzzy normal subsemirings of R is an (Q,L)-fuzzy normal subsemiring of R.
Proof: Let {A_{i}}_{i}_{ÎI} be a family of (Q,L)-fuzzy normal subsemirings of a semiring R and let A=. Then for x and y in R and q in Q. Clearly the intersection of a family of (Q,L)-fuzzy subsemirings of a semiring R is an (Q,L)-fuzzy subsemiring of a semiring R.(i) A(x+y,q) =A_{i}(x+y,q)=A_{i}(y+x,q)=A(y+x,q). Therefore, A(x+y,q)=A(y+x,q), for all x and y in R and q in Q. (ii) A(xy,q)=A_{i}(xy,q)= A_{i}(yx,q)=A(yx,q). Therefore, A(xy,q)=A(yx,q), for all x and y in R and q in Q.Hence the intersection of a family of (Q,L)-fuzzy normal subsemirings of a semiring R is an (Q,L)-fuzzy normal subsemiring of a semiring R.
3.3. Theorem 3
Let A and B be (Q,L)-fuzzy subsemiring of the semirings G and H, respectively. If A and B are (Q,L)-fuzzy normal subsemirings, then A×B is an (Q,L)- fuzzy normal subsemiring of G×H.
Proof: Let A and B be (Q,L)-fuzzy normal subsemirings of the semirings G and H respectively. Clearly A×B is an (Q,L)-fuzzy subsemiring of G×H. Let x_{1} and x_{2} be in G, y_{1} and y_{2} be in H and q in Q. Then (x_{1},y_{1}) and (x_{2},y_{2}) are in G×H. Now,A×B[(x_{1},y_{1})+(x_{2},y_{2}),q]=A×B((x_{1}+x_{2},y_{1}+y_{2}),q) =A(x_{1}+x_{2},q)˄B(y_{1}+y_{2},q)=A(x_{2}+x_{1},q)˄B(y_{2}+y_{1},q) =A×B((x_{2}+x_{1},y_{2}+y_{1}),q)=A×B[(x_{2},y_{2})+(x_{1},y_{1}),q]. Therefore,A×B[(x_{1},y_{1})+(x_{2},y_{2}),q]=A×B[(x_{2},y_{2})+(x_{1},y_{1}),q]. And,A×B[(x_{1},y_{1})(x_{2},y_{2}),q]=A×B((x_{1}x_{2},y_{1}y_{2}),q) =A(x_{1}x_{2},q)˄B(y_{1}y_{2},q)=A(x_{2}x_{1},q),B(y_{2}y_{1},q)= A×B((x_{2}x_{1},y_{2}y_{1}),q)=A×B[(x_{2},y_{2})(x_{1},y_{1}),q]. Therefore, A×B[(x_{1},y_{1})(x_{2},y_{2}),q]=A×B[(x_{2},y_{2})(x_{1},y_{1}),q]. Hence A×B is an (Q,L)-fuzzy normal subsemiring of G×H.
3.4. Theorem 4
Let A be a fuzzy subset in a semiring R and V be the strongest (Q,L)-fuzzy relation on R. Then A is an (Q,L)-fuzzy normal subsemiring of R if and only if V is an (Q,L)-fuzzy normal subsemiring of R×R.
Proof: Suppose that A is a (Q,L)-fuzzy normal subsemiring of R. Then for any x=(x_{1},x_{2}) and y=(y_{1},y_{2}) are in R×R and q in Q. Clearly V is a (Q,L)-fuzzy subsemiring of R×R. We have, V(x+y,q)=V[(x_{1},x_{2})+(y_{1},y_{2}),q]=V((x_{1}+y_{1},x_{2}+y_{2}),q) =A((x_{1}+y_{1}),q)ÙA((x_{2}+y_{2}),q)=A((y_{1}+x_{1}),q)ÙA((y_{2}+x_{2}),q) =V((y_{1}+x_{1},y_{2}+x_{2}),q)=V[(y_{1},y_{2})+(x_{1},x_{2}),q]=V(y+x,q) Therefore,V(x+y,q)=V(y+x,q), for all x and y in R×R and q in Q. We have, V(xy,q)=V[(x_{1},x_{2})(y_{1},y_{2}),q]=V((x_{1}y_{1},x_{2}y_{2}),q) =A((x_{1}y_{1}),q)ÙA((x_{2}y_{2}),q)=A((y_{1}x_{1}),q)ÙA((y_{2}x_{2}),q) =V((y_{1}x_{1},y_{2}x_{2}),q)=V[(y_{1},y_{2})(x_{1},x_{2}),q]=V(yx,q) Therefore, V(xy,q)=V(yx,q), for all x and y in R×R and q in Q. This proves that V is a (Q,L)-fuzzy normal subsemiring of R×R. Conversely, assume that V is a (Q,L)-fuzzy normal subsemiring of R×R, then for any x=(x_{1},x_{2}) and y=(y_{1},y_{2}) are in R×R,we have A(x_{1}+y_{1},q)ÙA(x_{2}+y_{2},q)=V((x_{1}+y_{1},x_{2}+y_{2}),q) =V[(x_{1},x_{2})+(y_{1},y_{2}),q]=V(x+y,q)=V(y+x,q) =V[(y_{1},y_{2})+(x_{1},x_{2}),q]=V((y_{1}+x_{1},y_{2}+x_{2}),q) =A(y_{1}+x_{1},q)ÙA(y_{2}+x_{2},q). We get, A((x_{1}+y_{1}),q)=A((y_{1}+ x_{1}),q), for all x_{1 }and y_{1} in R and q in Q. And A(x_{1}y_{1},q)ÙA(x_{2}y_{2},q)=V((x_{1}y_{1},x_{2}y_{2}),q)=V[(x_{1},x_{2})(y_{1},y_{2}),q] =V(xy,q)=V(yx,q)=V[(y_{1},y_{2})(x_{1},x_{2}),q]=V((y_{1}x_{1},y_{2}x_{2}),q) =A(y_{1}x_{1},q)Ù A(y_{2}x_{2},q). We get, A(( x_{1}y_{1}),q)=A((y_{1}x_{1}),q), for all x_{1 }and y_{1} in R and q in Q. Hence A is a (Q, L)-fuzzy normal subsemiring of R.
3.5. Theorem 5
Let (R,+, .) and (R^{׀},+, .) be any two semirings and Q be a non-empty set. The homomorphic image of an (Q,L)-fuzzy normal subsemiring of R is an (Q,L)-fuzzy normal subsemiring of R^{׀}.
Proof: Let (R,+,.) and (R^{׀},+,.) be any two semirings Q be a non-empty set and f :R®R^{׀ }be a homomorphism. Then, f(x+y)=f(x)+f(y) and f(xy)=f(x)f(y), for all x and y in R. Let V=f(A), where A is an (Q,L)-fuzzy normal subsemiring of a semiring R. We have to prove that V is an (Q,L)-fuzzy normal subsemiring of a semiring R^{׀}. Now, for f(x), f(y) in R^{׀}, clearly V is an (Q,L)-fuzzy subsemiring of a semiring R^{׀}, since A is an (Q,L)-fuzzy subsemiring of a semiring R. Now, V(f(x)+f(y),q)=V(f(x+y),q)≥A(x+y,q)=A(y+x,q)≤V(f(y+x),q)=V(f(y)+f(x),q), which implies that V(f(x)+f(y),q)= V(f(y)+(f(x),q), for all f(x) and f(y) in R^{׀}. Again, V(f(x)f(y),q)=V(f(xy),q)≥A(xy,q)=A(yx,q)≤V(f(yx),q) =V(f(y)f(x),q), which implies that V(f(x)f(y),q)=V(f(y)f(x),q ), for all f(x) and f(y) in R^{׀}. Hence V is an (Q,L)-fuzzy normal subsemiring of a semiring R^{׀}.
3.6. Theorem 6
Let (R,+,.) and (R^{׀},+,.) be any two semirings and Q be a non-empty set. The homomorphic preimage of an (Q,L)-fuzzy normal subsemiring of R^{׀} is an (Q,L)-fuzzy normal subsemiring of R.
Proof: Let ( R, +, .) and ( R^{׀}, +, .) be any two semirings and Q be a non-empty set and f : R ® R^{׀} be a homomorphism. Then, f(x+y)=f(x)+f(y) and f(xy)=f(x)f(y), for all x and y in R. Let V=f(A), where V is an (Q,L)-fuzzy normal subsemiring of a semiring R^{׀}. We have to prove that A is an (Q,L)-fuzzy normal subsemirring of a semiring R. Let x and y in R. Then, clearly A is an (Q,L)-fuzzy subsemiring of a semiring R, since V is an (Q,L)-fuzzy subsemiring of a semiring R^{׀}. Now, A(x+y,q)=V(f(x+y),q)=V(f(x)+f(y),q)=V(f(y)+f(x),q)= V(f(y+x),q)=A(y+x,q), which implies that A(x+y,q)=A(y+x,q), for all x and y in R and q in Q. Again, A(xy,q)=V(f(xy),q)=V(f(x)f(y),q)=V(f(y)f(x),q)=V(f(yx),q)=A(yx,q), which implies that A(xy,q)= A(yx,q), for all x and y in R and q in Q. Hence A is an (Q,L)-fuzzy normal subsemiring of a semiring R.
3.7. Theorem 7
Let (R, +, .) and (R^{׀}, +, .) be any two semirings and Q be a non-empty set. The anti-homomorphic image of an (Q, L)-fuzzy normal subsemiring of R is an (Q, L)-fuzzy normal subsemiring of R^{׀}.
Proof: Let (R,+,.) and (R^{׀},+,.) be any two semirings and Q be a non-empty set and f:R®R^{׀} be an anti-homomorphism. Then, f(x+y)=f(y)+f(x) and f(xy)=f(y) f(x), for all x and y in R. Let V=f(A), where A is an (Q,L)-fuzzy normal subsemiring of a semiring R. We have to prove that V is an (Q,L)-fuzzy normal subsemiring of a semiring R^{׀}. Now, for f(x) and f(y) in R^{׀}, clearly V is an (Q,L)-fuzzy subsemiring of a semiring R^{׀}, since A is an (Q,L)-fuzzy subsemiring of a semiring R. Now, V(f(x)+f(y),q)=V(f(y+x),q)≥A(y+x,q)=A(x+y,q)≤V(f(x+y),q)=V(f(y)+f(x),q), which implies that V(f(x)+f(y),q)=V(f(y)+f(x),q), for all f(x) and f(y) in R^{׀}. Again,V(f(x)f(y),q)=V(f(yx),q)≥A(yx,q)=A(xy,q)≤V(f(xy),q)=V(f(y)f(x),q), which implies that V(f(x)f(y),q)=V(f(y)f(x),q), for all f(x) and f(y) in R^{׀}. Hence V is an (Q,L)-fuzzy normal subsemiring of a semiring R^{׀}.
3.8. Theorem 8
Let (R,+,.) and (R^{׀},+,.) be any two semirings and Q be a non-empty set. The anti-homomorphic preimage of an (Q,L)-fuzzy normal subsemiring of R^{׀} is an (Q,L)-fuzzy normal subsemiring of R.
Proof: Let ( R, +, .) and ( R^{׀}, +, .) be any two semirings and Q be a non-empty set and f :R® R^{׀} be an anti-homomorphism. Then, f(x+y)=f(y)+f(x) and f(xy)=f(y)f(x), for all x and y in R.Let V=f(A), where V is an (Q,L)-fuzzy normal subsemiring of a semiring R^{׀}. We have to prove that A is an (Q,L)-fuzzy normal subsemiring of a semiring R. Let x and y in R, then clearly A is an (Q,L)-fuzzy subsemiring of a semiring R, since V is an (Q,L)-fuzzy subsemiring of a semiring R^{׀}. Now, A(x+ y,q)=V(f(x+y),q)=V(f(y)+f(x),q)=V(f(x)+f(y),q)=V(f(y+x),q)=A(y+x,q), which implies that A(x+y,q)= A(y+x,q), for all x and y in R and q in Q. Again, A(xy,q)=V(f(xy),q)=V(f(y)f(x),q)=V(f(x)f(y),q)=V(f(yx),q)=A(yx,q), which implies that A(xy,q)=A(yx,q), for all x and y in R and q in Q. Hence A is an (Q,L)-fuzzy normal subsemiring of a semiring R.
3.9. Theorem 9
Let A be an (Q, L)-fuzzy normal subsemiring of a semiring (R, +, .), then the pseudo (Q, L)-fuzzy coset (aA)^{p} is an (Q, L)-fuzzy normal subsemiring of a semiring R, for a in Rand q in Q.
Proof: Let A be an (Q, L)-fuzzy normal subsemiring of a semiring R. For every x and y in Rand q in Q, we have, ((aA)^{p})(x+y)=p(a)A(x+y)≥p(a){(A(x)˄A(y)}= {p(a)A(x)˄p(a)A(y)}={((aA)^{p})(x)˄((aA)^{p})(y)}. Therefore, ((aA)^{p})(x+y)={((aA)^{p})(x)˄((aA)^{p})(y)}. Now, ((aA)^{p})(xy)=p(a)A(xy)≥p(a){A(x)˄A(y)} = {p(a)A(x)˄p(a)A(y)}={((aA)^{p})(x)˄((aA)^{p})(y)}. Therefore, ((aA)^{p})(xy)={((aA)^{p})(x)˄((aA)^{p})(y)}. Hence (aA)^{p} is an (Q, L)-fuzzy normal subsemiring of a semiring R.
3.10. Theorem 10
Let A and B be (Q,L)-fuzzy subsets of the sets R and H respectively , and let α in L. Then (A×B)_{α} =A_{α }× B_{α}.
Proof: Let α in L. Let (x,y) be in (A×B)_{α} if and only if A×B( (x,y),q) ≥ α
if and only if {A(x,q)˄B(x,q)} ≥ α
if and only if A(x,q)≥α and B(x,q) ≥ α
if and only if xϵ A_{α }and yϵ B_{α}
if and only if (x,y) ϵ A_{α }× B_{α.}
Therefore, (A×B)_{α} =A_{α }× B_{α}.
3.11. Theorem 11
Let A be a (Q,L)-fuzzy normal subsemiring of a semiring R. If A(x,q) < A(y,q), for some x and y in R and q in Q, then A(x+y,q)=A(x,q)= A(y+x,q), for some x and y in R and q in Q.
Proof: It is trivial.
3.12. Theorem 12
Let A be a (Q,L)-fuzzy normal subsemiring of a semiring R. If A(x,q) > A(y,q), for some x and y in R and q in Q, then A(x+y,q)=A(y,q)= A(y+x,q), for some x and y in R and q in Q.
Proof: It is trivial.
3.13. Theorem 13
Let A be a (Q,L)-fuzzy normal subsemiring of a semiring R such that Im A ={α}, where α in L. If A=BC, where B and C are (Q,L)-fuzzy normal subsemiring of a semiring R, then either BC or CB.
Proof: It is trivial.
4. In the Following Theorem is the Composition Operation of Functions
4.1. Theorem 1
Let A be an (Q, L)-fuzzy normal subsemiring of a semiring H and f is an isomorphism from a semiring R onto H. Then A◦f is an (Q,L)-fuzzy normal subsemiring of the semiring R.
Proof: Let x and y in R and A be an (Q,L)-fuzzy normal subsemiring of a semiring H. Then clearly A◦f is an (Q,L)-fuzzy subsemiring of a semiring R. Now, (A◦f)( x+y, q) =A(f(x+y),q)=A(f(x)+f(y),q)=A(f(y)+f(x),q)=A(f(y+x),q) =(A◦f)(y+x,q),which implies that (A◦f)(x+y,q)=(A◦f)(y+x,q) , for all x and y in R and q in Q. And, (A◦f)(xy,q)= A(f(xy),q)=A(f(x)f(y),q)=A(f(y)f(x),q)=A(f(yx),q)= (A◦f)(yx,q), which implies that (A◦f)(xy,q)=(A◦f)(yx,q) , for all x and y in R and q in Q. Hence A◦f is an (Q,L)-fuzzy normal subsemiring of a semiring R.
4.2. Theorem 2
Let A be an (Q,L)-fuzzy normal subsemiring of a semiring H and f is an anti-isomorphism from a semiring R onto H. Then A◦f is an (Q,L)-fuzzy normal subsemiring of the semiring R.
Proof: Let x and y in R and A be an (Q,L)-fuzzy normal subsemiring of a semiring H. Then clearly A◦f is an (Q,L)-fuzzy subsemiring of a semiring R. Now, (A◦f)(x+y,q) = A(f(x+y),q)=A(f(y)+f(x),q)=A(f(x)+f(y),q)=A(f(y+x),q)= (A◦f)(y+x,q),which implies that (A◦f)(x+y,q)=(A◦f)(y+x,q) , for all x and y in R and q in Q. And, (A◦f)(xy,q) = A(f(xy),q)=A(f(y)f(x),q)=A(f(x)f(y),q)=A(f(yx),q)= (A◦f)(yx,q), which implies that (A◦f)(xy,q)=(A◦f)(yx,q) , for all x and y in R and q in Q. Hence A◦f is an (Q,L)-fuzzy normal subsemiring of a semiring R.
Acknowledgements
The authors would like to be thankful to the anonymous reviewers for their valuable suggestions.
References