Let $X$ be a set. The full transformation semigroup $T_X$ is the set of all maps from $X$ into $X$ with composition of maps as binary operation. A map $\alpha \in T_X$, is a subset of the cartesian product $X \times X$. A partial map $\beta$ is a map from a subset $Y \subset X$ into $X$ and it is also a subset of $X \times X$. We'll call $P_X$, the set of all partial maps of subsets $Y \subset X$ into $X$. Let $\beta, \gamma \in P_X$ be two partial maps. Then regular definition of maps composition doesn't work with $\beta$ and $\gamma$ because in general $\operatorname{dom} \beta \not = \operatorname{im} \gamma$ and $\operatorname{dom} \gamma \not = \operatorname{im} \beta$. However if one uses composition of binary relations as defined in J. M. Howie, Fundamentals of Semigroup Theory, 1995 (eq. 1.4.2 p.16)
$$\rho \circ \sigma = \{ (x,y) \in X \times X : (\exists z \in X) (x,z) \in \rho \text{ and } (z,y) \in \sigma \}$$
then $P_X$ can be equipped with this operation to form a semigroup. Lets just call this semigroup $P_X$ and take a partial map $\beta \in P_X$. We have $\beta^1 = \beta$, $\beta^2 = \beta \circ \beta$ and more generally $\beta^n = \beta \circ \beta^{n-1}$. For $\beta^n = \beta^m$, $n \not = m$, $\beta^n$ is either the empty set $\emptyset$, an idempotent element $\not = \emptyset$ or an element generating a subgroup of $P_X$ with order strictly greater than 1.
My question concerns the case where $\beta^n = \emptyset$. I would be tempted to call $\beta$ a nilpotent partial map. Is it the usual name given to it in the litterature or does a nilpotent partial map refers usually to something different? References on this topic would also be welcomed.
A preliminary remark. The set $F_n$ of all partial maps on $\{1, \dots, n\}$ is a monoid under the composition of partial maps. It can be embedded into $T_{n+1}$ by completing a partial function $f \in F_n$ to a transformation $\hat f$ on $\{0, 1, \dots, n\}$ by setting $\hat f(0) = 0$ and, for $x \in \{1, \dots, n\}$, $\hat f(x) = 0$ if $x$ is not in the domain of $f$.
Anyway, $F_n$ is a semigroup with zero. An element $s$ of a semigroup with zero is nilpotent if some power of $s$ is equal to $0$. Thus the term nilpotent partial map is perfectly legitimate in my opinion.