Abstract
In this study, we introduce the concepts of fuzzy subalgebras and fuzzy ideals in TM-algebras and investigate some of its properties. Problem statement: Let X be a TM-algebra, S be a subalgebra of X and I be a T-ideal of X. Let µ and v be fuzzy sets in a TM-algebra X. Approach: Define the upper level subset µt of µ and the cartesian product of µ and v from X×X to [0,1] by minimum of µ (x) and v (y) for all elements (x, y) in X×X. Result: We proved any subalgebra of a TM-algebra X can be realized as a level subalgebra of some fuzzy subalgebra of X and µt is a T-ideal of X. Also we proved, the cartesian product of µ and v is a fuzzy T-ideal of X×X. Conclusion: In this article, we have fuzzified the subalgebra and ideal of TM-algebras into fuzzy subalgrbra and fuzzy ideal of TM-algebras. It has been observed that the TM-algebra satisfy the various conditions stated in the BCC/ BCK algebras. These concepts can further be generalized.
Highlights
Isaki and Tanaka introduced two classes of abstract algebras BCI-algebras and BCK-algebras
We introduce the concepts of fuzzy subalgebras and fuzzy T-ideals in TM-algebra and investigate some of their properties
Definition 9: A fuzzy set μ in a TM-algebra X is called a fuzzy subalgebra of X if μ(x * y) ≥ min{μ(x),μ(y)},for all x, y ∈ X
Summary
Isaki and Tanaka introduced two classes of abstract algebras BCI-algebras and BCK-algebras. We introduce the concepts of fuzzy subalgebras and fuzzy T-ideals in TM-algebra and investigate some of their properties. Theorem 14: A fuzzy set μ of a TM-algebra X is a fuzzy subalgebra if and only if for every t ∈[0,1] , μt is either empty or a subalgebra of X. Definition 7: Let (X,*,0) be a TM-algebra.
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