Communications of the
Korean Mathematical Society CKMS

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

Keywords: hyper $K$-algebra, fuzzy hyper $K$-subalgebra (of type $(\alpha,\beta)$), support, $(\in,$ $\in\! \vee \, {\rm q})$-fuzzy hyper $K$-subalgebra, fuzzifying hyper $K$-subalgebra, implication-based fuzzy hyper $K$-subalgebra

MSC numbers: 06F35, 03G25, 08A72