Natural Partial Order on the Semigroups of Partial Isometries of a Finite Chain
Chaiwat Namnak, Ekkachai Laysirikul, Nares Sawatraksa · Thai Journal of Mathematics · 2018
Let $I_n$ denote the $1-1$ partial transformation semigroup on a set $\{1, 2, \ldots, n\}$ and let $DP_n = \{\alpha \in I_n : \forall x, y \in Dom \alpha , |x\alpha -y\alpha|=|x-y|\}$ and $ ODP_n = \{\alpha \in DP_n : \forall x,y \in Dom\,\alpha, x \leq y \Rightarrow x\alpha \leq y\alpha\}$. Then $DP_n$ and $ODP_n$ are subsemigroups of $I_n$. The purpose of this research, we study the natural partial orders on $DP_n$ and $ODP_n$ and characterize when two elements of $DP_n$ and $ODP_n$ are related under this partial order. Moreover, we give a necessary and sufficient conditions for elements in $DP_n$ and $ODP_n$ to be maximal or minimal elements.