• Journal of the Chungcheong Mathematical Society
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• v.33 no.4
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• pp.395-404
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• 2020

Based on a sub-BCK-algebra K of a BCI-algebra X, the notions of fuzzy (K, ε)-subalgebras, fuzzy (K, ε)-ideals and fuzzy commutative (K, ε)-ideals are introduced, and their relations/properties are investigated. Conditions for a fuzzy subalgebra/ideal to be a fuzzy (K, ε)-subalgebra/ideal are provided.

• Bulletin of the Korean Mathematical Society
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• v.34 no.2
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• pp.205-209
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• 1997

Hong et al. [2] and Jun et al. [4] introduced the notion of k-nil radical in a BCI-algebra, and investigated its some properties. In this paper, we discuss the further properties on the k-nil radical. Let A be a subset of a BCI-algebra X. We show that the k-nil radical of A is the union of branches. We prove that if A is an ideal then the k-nil radical [A;k] is a p-ideal of X, and that if A is a subalgebra, then the k-nil radical [A;k] is a closed p-ideal, and hence a strong ideal of X.

• Journal of the Chungcheong Mathematical Society
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• v.9 no.1
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• pp.11-15
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• 1996

In this paper we prove that if Y is a poset of the form $\underline{1}{\oplus}Y^{\prime}$ for some subposet Y' then BCK(Y) is a ${\Gamma}$-BCK-algebra. Moreover, if X is a BCI-algebra then Hom(X, BCK(Y)) is a positive implicative ${\Gamma}$-BCK-algebra.

• Honam Mathematical Journal
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• v.36 no.3
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• pp.623-635
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• 2014

Classications of (${\alpha},{\beta}$)-fuzzy subalgebras of BCK/BCI-algebras are discussed. Relations between (${\in},{\in}{\vee}q$)-fuzzy subalgebras and ($q,{\in}{\vee}q$)-fuzzy subalgebras are established. Given special sets, so called t-q-set and t-${\in}{\vee}q$-set, conditions for the t-q-set and t-${\in}{\vee}q$-set to be subalgebras are considered. The notions of $({\in},q)^{max}$-fuzzy subalgebra, $(q,{\in})^{max}$-fuzzy subalgebra and $(q,{\in}{\vee}q)^{max}$-fuzzy subalgebra are introduced. Conditions for a fuzzy set to be an $({\in},q)^{max}$-fuzzy subalgebra, a $(q,{\in})^{max}$-fuzzy subalgebra and a $(q,{\in}{\vee}q)^{max}$-fuzzy subalgebra are considered.

• Korean Journal of Mathematics
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• v.24 no.3
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• pp.345-367
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• 2016

The concept of derivations of BCI-algebras was first introduced by Jun and Xin. In this paper, we introduce the notions of (l, r)-derivations, (r, l)-derivations and derivations of UP-algebras and investigate some related properties. In addition, we define two subsets $Ker_d(A)$ and $Fix_d(A)$ for some derivation d of a UP-algebra A, and we consider some properties of these as well.

• Bulletin of the Korean Mathematical Society
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• v.42 no.4
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• pp.703-711
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• 2005

Using the belongs to relation ($\in$) and quasi-coincidence with relation (q) between fuzzy points and fuzzy sets, the concept of (${\alpha},\;{\beta}$)-fuzzy subalgebras where ${\alpha},\;{\beta}$ are any two of $\{\in,\;q,\;{\in}\;{\vee}\;q,\;{\in}\;{\wedge}\;q\}$ with $\;{\alpha}\;{\neq}\;{\in}\;{\wedge}\;q$ is introduced, and related properties are investigated.

• Journal of applied mathematics & informatics
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• v.11 no.1_2
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• pp.255-263
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• 2003

Fuzzifications of n-fold quasi-associative ideals are considered. Conditions for a fuzzy ideal to be a fuzzy n-fold quasi-associative ideal are given. Using a collection of n-fold quasi-associative ideals, fuzzy n-fold quasi-associative ideals are constructed. Finally, the extension property for fuzzy n-fold quasi-associative ideals is established.

• Kyungpook Mathematical Journal
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• v.56 no.2
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• pp.371-386
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• 2016

Using the hesitant intersection (${\Cap}$), the notions of ${\Cap}$-hesitant fuzzy subalgebras, ${\Cap}$-hesitant fuzzy ideals and ${\Cap}$-hesitant fuzzy p-ideals are introduced,and their relations and related properties are investigated. Conditions for a ${\Cap}$-hesitant fuzzy ideal to be a ${\Cap}$-hesitant fuzzy p-ideal are provided. The extension property for ${\Cap}$-hesitant fuzzy p-ideals is established.

• Communications of the Korean Mathematical Society
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• v.15 no.3
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• pp.499-509
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• 2000

In [9], the concept of fuzzy sets is applied to the theory of Ζ-ideals in a BCI-semigroup (it was renamed as an IS-algebra for the convenience of study), and a characterization of fuzzy Ζ-ideals by their level Ζ-ideals was discussed. In this paper, we study further properties of fuzzy Ζ-ideals. We prove that the homomorphic image and preimage of a fuzzy Ζ-ideal are also fuzzy Ζ-ideals.

• Kyungpook Mathematical Journal
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• v.47 no.3
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• pp.403-410
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• 2007

Using the belongs to relation (${\in}$) and quasi-coincidence with relation (q) between fuzzy points and fuzzy sets, the concept of (${\alpha}$, ${\beta}$)-fuzzy subalgebras where ${\alpha}$ and ${\beta}$ areany two of {${\in}$, q, ${\in}{\vee}q$, ${\in}{\wedge}q$} with ${\alpha}{\neq}{\in}{\wedge}q$ was already introduced, and related properties were investigated (see [3]). In this paper, we give a condition for an (${\in}$, ${\in}{\vee}q$)-fuzzy subalgebra to be an (${\in}$, ${\in}$)-fuzzy subalgebra. We provide characterizations of an (${\in}$, ${\in}{\vee}q$)-fuzzy subalgebra. We show that a proper (${\in}$, ${\in}$)-fuzzy subalgebra $\mathfrak{A}$ of X with additional conditions can be expressed as the union of two proper non-equivalent (${\in}$, ${\in}$)-fuzzy subalgebras of X. We also prove that if $\mathfrak{A}$ is a proper (${\in}$, ${\in}{\vee}q$)-fuzzy subalgebra of a CK/BCI-algebra X such that #($\mathfrak{A}(x){\mid}\mathfrak{A}(x)$ < 0.5} ${\geq}2$, then there exist two prope non-equivalent (${\in}$, ${\in}{\vee}q$)-fuzzy subalgebras of X such that $\mathfrak{A}$ can be expressed as the union of them.