Generalized Fuzzy BCK-algebra And Generalized Fuzzy Ring

Based on the concepts of fuzzy BCK-algebra, fuzzy BCH-algebra, we study the broader (λ₁ ,λ₂ )- generalized fuzzy sub-algebra, (∈,∈∨q_{λ1 ,λ2})- fuzzy sub-algebra on BCK(BCH)– algebra and several kinds of (λ₁ ,λ₂ )- generalized fuzzy ideal and(∈,∈∨q_{λ1 ,λ2})- fuzzy ideal, then we get some important properties. In the end, (∈,∈∨q_λ,μ)- fuzzy subring, (∈,∈∨q_λ,μ)- fuzzy ideal on the ring and (∈,∈∨q_λ,μ)- fuzzy(completely)regular subring are studied. The main results are listed as follows:(1)Intoduce the concepts of (λ₁ ,λ₂ )-generalized fuzzy subalgebra of BCK-algebra,(λ₁ ,λ₂ )-generalized fuzzy ideal, (λ₁ ,λ₂ )- generalized fuzzy implicative ideal, then discuss some properties of (λ₁ ,λ₂ )-generalized fuzzy subalgebra by its level setμαand characteristic function f Y, also we prove that when"λ1 = 0,λ2= 1", fuzzy BCK-algebra in common sense is a special case of (λ₁ ,λ₂ )-generalized fuzzy BCK-algebra. Then by"belong to in the generalization"the author gives the concepts of (∈,∈∨q_{λ1 ,λ2})-fuzzy subalgebra,(∈,∈∨q_{λ1 ,λ2})-fuzzy ideal,(∈,∈∨q_{λ1 ,λ2})-fuzzy implicative ideal,Whenλ=0,μ=0.5",belong to"in the generalization is equal to"belong to"in the common sense. Then we give three equal depictions: among (λ₁ ,λ₂ )-generalized fuzzy subalgebra ,(∈,∈∨q_{λ1 ,λ2})-fuzzy subalgebra,a non-empty level setμα; among (λ₁ ,λ₂ )-generalized fuzzy ideal,(∈,∈∨q_{λ1 ,λ2})- fuzzy ideal,a non-empty idealμαon BCK-algebra; among (λ₁ ,λ₂ )-generalized fuzzy implicative ideal,(∈,∈∨q_{λ1 ,λ2})-fuzzy implicative ideal,a non-empty implicative idealμαon BCK-algebra. In order to show those concepts more clearly, the author gives some examples. In the end, we discuss some basic properties using the concepts of homomorphic on BCK-algebra and supremum.(2)Using the similar studying method of BCK-algebra, on BCH-algebra, the author studies (λ₁ ,λ₂ )-generalized fuzzy ideal, (λ₁ ,λ₂ )-generalized fuzzy closed ideal,(∈,∈∨q_{λ1 ,λ2}) -fuzzy ideal,(∈,∈∨q_{λ1 ,λ2})-fuzzy closed ideal and also gives several equal depictions which are similar to those on BCK-algebra.(3) Introduce the concepts of (∈,∈∨q_λ,μ)- fuzzy rings,(∈,∈∨q_λ,μ)- fuzzy ideal on the ring, (∈,∈∨q_λ,μ)- fuzzy(completely)regular ideal. Also we study (∈,∈∨q_λ,μ)- fuzzy complete regular subring using group inverse x #(for any x∈R,there exists y∈R,such that xyx = x; yxy = y ;xy = yx,then y is called group inverse of x )and discuss some properties. Then we give the definition of the generalized fuzzy left(right,bi-,interior)ideal and (∈,∈∨q_λ,μ)- fuzzy left(right,bi-,interior)idea on the ring,and prove that generalized fuzzy left(right,bi-,interior)ideal and (∈,∈∨q_λ,μ)- fuzzy left(right,bi-,interior)ideal are equal in value. Next, we give the definition of the generalized fuzzy (completely)regular subring and (∈,∈∨q(λ,μ))- fuzzy (completely)regular subring,and also prove that the generalized fuzzy (completely)regular subring and (∈,∈∨q(λ,μ))- fuzzy (completely)regular subring are equal in value.