A partial permutation $\pi$ is a permutation of the elements of some finite set $S$ with the location of all elements except those in some set $S'\subseteq S$ unknown. For example, a partial permutation of the subset $\{1,2,3\}$ to $\{1,2,3,4,5,6\}$ is $\pi=(\lozenge,\lozenge,3,\lozenge,1,2)$ where $\lozenge$ denotes a spot occupied by an unknown element.
Given two partial permutations $\pi$ and $\sigma$ for the same $S$ but disjoint $S'_\pi$ and $S'_\sigma$ as well as disjoint $\pi(S'_\pi)$ and $\sigma(S'_\sigma)$, we furthermore define their sum to be a partial permutation $\pi+\sigma$ such that $S'_{\pi+\sigma}=S'_\pi\cup S'_\sigma$ and $\pi+\sigma$ agrees with $\pi$ and $\sigma$ on the location of all elements the location is known of. A partial permutation $\pi$ is defined to be full if it accounts for the location of all elements, i.e. $S'_\pi=S$.
I would like to know if it is possible to generalize the parity of a permutation $\def\sgn{{\rm sgn}}\sgn(\pi)$ such that it agrees with the conventional parity of a permutation for full permutations and $\sgn(\pi+\sigma)=\sgn(\pi)\sgn(\sigma)$ for disjoint $S'_\pi$ and $S'_\sigma$.