Bulletin of the
Korean Mathematical Society BKMS

In this paper, we deal with a characterization of the posets with the property that every poset isometry of $\mathbb{F}^n_q$ fixing the origin is a linear map. We say such a poset to be {\it admitting the linearity of isometries}. We show that a poset $P$ admits the linearity of isometries over $\mathbb{F}^n_q$ if and only if $P$ is a disjoint sum of chains of cardinality $2$ or $1$ when $q=2$, or $P$ is an anti-chain otherwise.

Keywords: $P$-isometries, $P$-isometry group, poset metric

MSC numbers: 94B05