# A graph associated to a polygroup with respect to an automorphism

Document Type : Original Article

Author

Department of Mathematics, Faculty of Science, Payame Noor University, Tehran, Iran

Abstract

In this paper, we introduce and study, $\zeta^\alpha(P)$, the $\alpha$-center of a polygroup $(P, \cdot )$ with respect to an automorphism $\alpha$. Then we associate to $P$ a graph $\Gamma^\alpha_{P}$, whose vertices are elements of $P \setminus \zeta^\alpha(P)$ and $x$ connected to $y$ by an edge in case $x \cdot y \cdot \omega \neq y \cdot x^\alpha \cdot \omega$ or $y \cdot x \cdot \omega \neq x \cdot y^\alpha \cdot \omega$, where $\omega$ is the heart of $P$. We obtain some basic properties of this graph. In particular, we prove that if $\zeta^\alpha(P) \neq P$, then $dim(\Gamma^\alpha _{P})=2$. Moreover, we define a weak $\alpha$-commutative polygroup to state that if $\Gamma^\alpha_{H} \cong \Gamma^\beta_{K}$ and $H$ is a weak $\alpha$-commutative, then $K$ is a weak $\beta$-commutative. Also, we show that if $H$ and $K$ are two polygroups such that $\Gamma^\alpha_{H} \cong \Gamma^\beta_{K}$, then for some automorphisms $\eta$ and $\lambda$, $\Gamma^\eta_{H \times A} \cong \Gamma^\lambda_{K \times B}$, where $A$ and $B$ are two weak commutative polygroups.

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