• Honam Mathematical Journal
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• v.42 no.2
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• pp.219-234
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• 2020

The fundamental relation on a fuzzy hyperring is defined as the smallest equivalence relation, such that the quotient would be the ring, that is not commutative necessarily. In this paper, we introduce a new fuzzy strongly regular equivalence on fuzzy hyperrings, where the ring is commutative with respect to both sum and product. With considering this relation on fuzzy hyperring, the set of the quotient is a commutative ring. Also, we introduce fundamental functor between the category of fuzzy hyperrings and category of commutative rings and some related properties. Eventually, we introduce α-part in fuzzy hyperring and determine some necessary and sufficient conditions so that the relation α is transitive.

• Korean Journal of Mathematics
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• v.6 no.2
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• pp.165-172
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• 1998

For a G-map ${\phi}:X{\rightarrow}X$, we define and characterize the Reidemeister number $R_G({\phi})$ of ${\phi}$. Also, we prove that $R_G({\phi})$ is a G-homotopy invariance and we obtain a lower bound of $R_G({\phi})$.

• Communications of the Korean Mathematical Society
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• v.11 no.2
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• pp.445-455
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• 1996

In this paper we study the Reidemeister number $R(f_G)$ for a self-map $f_G : (X, G) \to (X, G)$ of the transformation group (X,G), as an extenstion of the Reidemeister number R(f) for a self-map $f : X \to X$ of a topological space X.

• Korean Journal of Mathematics
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• v.5 no.2
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• pp.177-183
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• 1997

Let ${\sigma}(X,x_0,G)$ be the fundamental group of a transformation group (X,G). Let $R({\varphi},{\psi})$) be the generalized Reidemeister number for an endomorphism $({\varphi},{\psi}):(X,G){\rightarrow}(X,G)$. In this paper, our main results are as follows ; we prove some sufficient conditions for $R({\varphi},{\psi})$ to be the cardinality of $Coker(1-({\varphi},{\psi})_{\bar{\sigma}})$, where 1 is the identity isomorphism and $({\varphi},{\psi})_{\bar{\sigma}}$ is the endomorphism of ${\bar{\sigma}}(X,x_0,G)$, the quotient group of ${\sigma}(X,x_0,G)$ by the commutator subgroup $C({\sigma}(X,x_0,G))$, induced by (${\varphi},{\psi}$). In particular, we prove $R({\varphi},{\psi})={\mid}Coker(1-({\varphi},{\psi})_{\bar{\sigma}}){\mid}$, provided that (${\varphi},{\psi}$) is eventually commutative.