• Communications of the Korean Mathematical Society
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• v.22 no.2
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• pp.173-181
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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 $\alpha,\;\beta$ are any two of $\{{\in},\;q,\;{\in}\;{\vee}\;q,\;{\in}\;{\wedge}\;q\}$ with ${\alpha}\;{\neq}\;{\in}\;{\wedge}\;q$ was introduced, and related properties were investigated in [3]. As a continuation of the paper [3], in this paper, the notion of a fuzzy subalgebra with thresholds is introduced, and its characterizations are obtained. Relations between a fuzzy subalgebra with thresholds and an (${\in},\;{\in}\;{\vee}\;q$)-fuzzy subalgebra are provided.

• 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.

• Communications of the Korean Mathematical Society
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• v.26 no.3
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• pp.385-403
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• 2011

Generalizations of the notion of fuzzy hyper K-subalgebras are considered. The concept of fuzzy hyper K-subalgebras of type (${\alpha},{\beta}$) where ${\alpha}$, ${\beta}$ ${\in}$ {${\in}$, q, ${\in}{\vee}q$, ${\in}{\wedge}q$} and ${\alpha}{\neq}{\in}{\wedge}q$. Relations between each types are investigated, and many related properties are discussed. In particular, the notion of (${\in}$, ${\in}{\vee}q$)-fuzzy hyper K-subalgebras is dealt with, and characterizations of (${\in}$, ${\in}{\vee}q$)-fuzzy hyper K-subalgebras are established. Conditions for an (${\in}$, ${\in}{\vee}q$)-fuzzy hyper K-subalgebra to be an (${\in}$, ${\in}$)-fuzzy hyper K-subalgebra are provided. An (${\in}$, ${\in}{\vee}q$)-fuzzy hyper K-subalgebra by using a collection of hyper K-subalgebras is established. Finally the implication-based fuzzy hyper K-subalgebras are discussed.

• 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.