Can we generalize parity to partial permutations?

Question

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 $S'_\sigma$.

Answer 1

Here is a contradiction in the case of $3$ elements, following FUZxxl's answer post.

Let's say we can assign $\def\sgn{{\rm sgn}}\sgn(x_k, k)$ in a consistent manner. Let's put those assignments in a $3\times 3$ matrix where $a_{ij}$ is $\sgn(i, j)$. For this to give the correct sign to all full permutations, we would need the following relations to hold: $$ a_{11}a_{22}a_{33} = a_{12}a_{23}a_{31} = a_{13}a_{21}a_{32} = 1\\ a_{12}a_{21}a_{33} = a_{23}a_{32}a_{11} = a_{13}a_{31}a_{22} = -1 $$ I will show that this is, in fact, impossible.

I will divide into two cases: Either all of $a_{ii}$ are $1$, or exactly two of them are $-1$.

First case: The second relation above gives $a_{13} = -a_{31}, a_{23} = -a_{32}$ and $a_{12} =-a_{21}$. But inserting that into the first relation gives $a_{12}a_{23}(-a_{13}) = (-a_{12})(-a_{23})a_{13}$ which is a contradiction.

Second case: two of the $a_{ii}$ are $-1$. WLOG, assume that it's $a_{22}$ and $a_{33}$. The second relation then gives $a_{31} = a_{13}, a_{23} = -a_{32}$ and $a_{12} = a_{21}$. But then the first relation gives us $a_{12}a_{23}a_{13} = a_{13}a_{12}(-a_{23})$ which is again a contradiction.

Answer 2

I believe no such generalisation exists. Here is a proof idea by contradiction:

Suppose $\def\sgn{{\rm sgn}}\sgn(\pi)$ is such a parity such that $\sgn(\pi+\sigma)=\sgn(\pi)\sgn(\sigma)$. We can use this to represent the parity of a full permutation as a product of parities of partial permutations such that each partial permutation accounts for the location of a single element:

$$\sgn(x_1,x_2,\ldots,x_n)=\sgn(x_1,\lozenge,\ldots,\lozenge)\sgn(\lozenge,x_2,\ldots,\lozenge)\cdots\sgn(\lozenge,\lozenge,\ldots,x_n)$$

Let us now define

$$\sgn_n(x_k,k)=\sgn\underbrace{(\lozenge,\ldots,x_k,\ldots,\lozenge)}_{\mbox{$n$ elements}}$$

where $x_k$ is at location $k$ and we can write

$$\sgn(x_1,x_2,\ldots,x_n)=\sgn_n(x_1,1)\sgn_n(x_2,2)\cdots\sgn_n(x_n,n).$$

Now I'm pretty sure that the sign of a permutation cannot be expressed as a product of functions in the locations of individual elements, but I am not quite sure how to show that. My idea is to use that transposing the elements at indices $i$ and $j$ must flip the sign of exactly one of them but I'm not sure how to build a contradiction from that.