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Fuzzy and Neutrosophic Sets in Semigroups
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The notions of hyperfuzzy ideals in B C K / B C I -algebras are introduced, and related properties are investigated. Characterizations of hyperfuzzy ideals are established. Relations between hyperfuzzy ideals and hyperfuzzy subalgebras are discussed. Conditions for hyperfuzzy subalgebras to be hyperfuzzy ideals are provided.
In this paper, we deal with soft MTL-algebras based on fuzzy sets. By means
of $\in$-soft sets and q-soft sets, some characterizations of (Boolean, G- and
MV-) filteristic soft MTL-algebras are investigated. Finally, we prove that a
soft set is a Boolean filteristic soft MTL-algebra if and only if it is both a
G-filteristic soft MTL-algebra and an MV-filteristic soft MTL-algebra.
Molodtsov [D. Molodtsov, Soft set theory — First results, Comput. Math. Appl. 37 (1999) 19–31] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. In this paper we apply the notion of soft sets by Molodtsov to the theory of BCK/BCI-algebras. The notion of soft BCK/BCI-algebras and soft subalgebras are introduced, and their basic properties are derived.
The notion of soft ideals and idealistic soft BCK/BCI-algebras is introduced, and several examples are given. Relations between soft BCK/BCI-algebras and idealistic soft BCK/BCI-algebras are provided. The intersection, union, “AND” operation, and “OR” operation of soft ideals and idealistic soft BCK/BCI-algebras are established.
More general form of an ({\in},{\in}{\vee}q_k)-fuzzy subsemigroup is considered. The notions of ({\in},q^{\delta}_k)-fuzzy subsemigroup, (q^{\delta}_0,{\in}{\vee}q^{\delta}_k)-fuzzy subsemigroup and ({\in},{\in}{\vee}q^{\delta}_k)-fuzzy subsemigroup are introduced, and related properties are investigated. Characterizations of an ({\in},{\in}{\vee}q^{\delta}_k)-fuzzy subsemigroup are considered. Conditions for an ({\in},{\in}{\vee}q^{\delta}_k)-fuzzy subsemigroup to be a fuzzy subsemigroup are provided. Relations between (q^{\delta}_0,{\in}{\vee}q^{\delta}_k)-fuzzy subsemigroup, ({\in},q^{\delta}_k)-fuzzy subsemigroup and ({\in},{\in}{\vee}q^{\delta}_k)-fuzzy subsemigroup are discussed.
The notion of a rough fuzzy filter in a BE-algebra is introduced and some properties of such a filter are investigated. The relations between the upper (lower) rough filters and the upper (lower) approximations of their homomorphism images are discussed.
The notions of starshaped (∈, ∈ q)-fuzzy sets and quasi-starshaped (∈, ∈ q)-fuzzy sets are introduced, and related properties are investigated. Characterizations of starshaped (∈, ∈ q)-fuzzy sets and quasi-starshaped (∈, ∈ q)-fuzzy sets are discussed.
The aim of this paper is to introduce the notions of (is an element of, is an element of V q)-fuzzy p-ideals, (is an element of, is an element of V q)-fuzzy q-ideals and (is an element of, is an element of V q)-fuzzy a-ideals in BCI-algebras and to investigate some of their properties. Several characterization theorems for these generalized fuzzy ideals are proved and the relationship among these generalized fuzzy ideals of BCI-algebras is discussed. It is shown that a fuzzy set of a BCI-algebra is an (is an element of, is an element of V q)-fuzzy a-ideal if and only if it is both an (is an element of, is an element of V q)-fuzzy p-ideal and an (is an element of, is an element of V q)-fuzzy q-ideal. Finally, the concept of implication-based fuzzy a-ideals in BCI-algebras is introduced and, in particular, the implication operators in Lukasiewicz system of continuous-valued logic are discussed.
In BCK-algebras, the notion of commutative intersection-soft ideal is introduced, and related properties are investigated. Characterizations of a commutative intersection-soft ideal are considered. Conditions for an intersection-soft ideal to be commutative are provided. Extension property of a commutative intersection-soft ideal is established. The problem of classifying (commutative) intersection-soft ideals by their inclusive sets is solved.