Abstract
In this paper, some types of fuzzy filters of a strong Sheffer stroke non-associative MV-algebra (for short, strong Sheffer stroke NMV-algebra) are introduced. By presenting new properties of filters, we define a prime filter in this algebraic structure. Then (prime) fuzzy filters of a strong Sheffer stroke NMV-algebra are determined and some features are proved. Finally, we built quotient strong Sheffer stroke NMV-algebra by a fuzzy filter.
Highlights
It is shown that α ◦ h is a fuzzy filter of A and the quotient structures defined by the fuzzy filters α ◦ h and α are isomorphic, for strong Sheffer stroke NMV-algebras A and B, an epimorphism h between these algebras and a fuzzy filter α of B
We prove that a filter of a strong Sheffer stroke NMV-algebra is prime if and only if it is not contained by another filter of this algebraic structure, and examine some features of a prime filter
It is shown that a fuzzy subset αh of a strong Sheffer stroke NMV-algebra is a fuzzy filter defined by means of a fuzzy filter α and a surjective endomomorphism h on this algebra, and that αh = α if and only if h(αa) = αa whenever h is an automorphism on this algebra and a ∈ Im(α)
Summary
It is proved that a filter of a strong Sheffer stroke NMV-algebra is prime if and only if the quotient structure defined by the filter is totally ordered strong Sheffer stroke NMV-algebra and its cardinality is less than or equals to 2. It is shown that F is a (prime) filter of a strong Sheffer stroke NMV-algebra if and only if a fuzzy subset αF defined by F is a (prime) fuzzy filter of this algebraic structure. It is demonstrated that a strong Sheffer stroke NMV-algebra is totally ordered if and only if every fuzzy filter is prime if and only if the filter {1} is prime. A fuzzy filter α of a strong Sheffer stroke NMV-algebra is prime if and only if the quotient structure is a totally ordered strong Sheffer stroke NMV-algebra and its cardinality is less than or equals to 2. It is stated that the class of all fuzzy filters of a strong Sheffer stroke NMV-algebra forms a complete lattice since the interval [0, 1] is a complete lattice and has important properties
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