Extended Fuzzy $BCK$-subalgebras

Document Type : Research Paper


1 Department of Mathematics, Hubei University for Nationalities, Enshi, Hubei 445000, China

2 Department of Mathematics, Payame Noor University, Tehran, Iran

3 Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran


This  paper extends the notion of fuzzy $BCK$-subalgebras to fuzzy hyper $BCK$-subalgebras and defines an extended fuzzy $BCK$-subalgebras. This study  considers  a type of fuzzy hyper  $BCK$-ideals  in this  hyperstructure  and  describes   the  relationship  between hyper  $BCK$-ideals and  fuzzy hyper  $BCK$-ideals.  In fact, it tries to introduce   a  strongly  regular  relation  on hyper $BCK$-algebras. Moreover,  by using  the   fuzzy hyper  $BCK$-ideals, it defines a congruence  relation  on (weak commutative) hyper $BCK$-algebras  that under some conditions is strongly regular and  the quotient of any hyper $BCK$-algebra via this relation  is a $($hyper  $BCK$-algebra$)$ $BCK$-algebra.


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