A left almost semigroup, or $LA$-semigroup, is a groupoid $S$ satisfying the left invertive law \begin{equation} (ab)c = (cb)a, \end{equation} for every $a,b,c\in S$, [8]. Condition (1) is in fact a left Abel-Grassmann's law, [4], and notion $LA$-semi- group reminds of associativity. Since concerned structure is not associative in order to avoid confusion we shall use notion Abel-Grassmann's groupoids or $AG$-groupoids. In this paper we define relation which is a natural partial order relation on AG-band, $AG^*$ and $AG^{**}$-groupoids. Also we introduce the notion of $r$-cancelative $\pi$-inverse $AG^*$-groupoid and on this structure we consider the natural partial order and maximal elements.