# STRONG COMPATIBILITY IN CERTAIN QUASIGROUP NONUNIFORM HOMOGENEOUS SPACES OF DEGREE 4

• Im, Bokhee (Department of Mathematics, Chonnam National University) ;
• Ryu, Ji-Young (Department of Mathematics, Chonnam National University)
• Accepted : 2019.04.23
• Published : 2019.09.25
• 81 18

#### Abstract

We consider quasigroups $Q({\Gamma})$ obtained as certain double covers of the symmetric group $S_3$ of degree 3, for directed graphs ${\Gamma}$ on the vertex set $S_3$. We completely characterize the strong compatibility of elements of $Q({\Gamma})$ for any quasigroup nonuniform homogeneous space of degree 4. For such homogeneous spaces, we classify all the strong and weak compatibility graphs of $Q({\Gamma})$.

#### Keywords

quasigroup;quasigroup action;homogeneous space;intercalate;strongly compatible;compatibility graph;compatibility

#### Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

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