The APL gradeup (gradedown) function (denoted by $\uparrow \{respectively\downarrow \}$) aapplied to a vector $v\in Rn$ grades' the elements of $v$ in ascending (descending) order. (Among equal elements of $v$ the ranking is determined by their position). For example, if $v=(2.3,4.7,6.8,0.6,3.7,4.7)$ then $\uparrow v$ is (4,1,5,2,6,3) and $\downarrow v$ is (3,2,6,5,1,4). An immediate consequence of the versatility of this functional form is that the expression $v[\uparrow v] (v[\downarrow v])$ sorts' the elements of $v$ in ascending (descending) order. The question of characterizing all vectors $v\in Rn$ such that $\uparrow v=v$ was answered in a paper by Cooper, Best and Kennedy [see Cooper, et al.(1)]. The results there are essentially determined by scrutinizing the selection property of $\uparrow v$; that is, if $v=(x1,x2,\dots ,xn)$ then $\uparrow v$ can be visualized as the unique permutation $P\in Sn$ such that $x\_\{P(1)\},x\_\{P(2),\backslash ldots,x\_\{P(n)$ is a list of the elements f $v$ in ascending order. Two straightforward consequences of this interpretation are: 1) if $P\in Sn$, then $\uparrow P$ is the inverse of $P$ and 2) the `fixed' points of the mapping $\uparrow$ are precisely the involutions of $Sn$. IN this note we continue this investigation for the gradedown function and also resolve the open questions posed in that paper.