The map $~ sgn^{\sharp}: {\cal SG}_{n} \to \{ +1, -1 \}$ [*is not a groupoid homomorphism*]{}.

Indeed, for $ X = \{ x_{1}, x_{2}, x_{3}, x_{4} \} $ we consider the permutations $ f, g \in {\cal SG}_{4} $, where $ f = \left (
\begin{array}{ccc} x_{1} & x_{2} &
x_{3}\\
x_{2} & x_{3} & x_{1}\\
\end{array}\right ) $ and $ g = \left ( \begin{array}{ccc} x_{1} & x_{3} &
x_{4}\\
x_{4} & x_{3} & x_{1}\\
\end{array}\right ).$ Then\
$~ sgn^{\sharp}(f) = + 1, ~ sgn^{\sharp}(g) = -1 $ and $~( sgn^{\sharp}(f), sgn^{\sharp}(g))\in \{ +1, -1\}\times \{ +1, -1\}
$. But $ f $ and $g$ are not composable in $ {\cal SG}_{4} $, since $ D(f)\neq D(g).$

VECTOR GROUPOIDS
================

A [**vector groupoid over a field $K$**]{}, is a groupoid $~(V,
\alpha, \beta, \odot, \iota,  V_0)$ such that:

(3.1.1) $~V$ is a vector space over $K$, and the units set $V_0$ is a subspace of $V$.

(3.1.2) The source and the target maps $\alpha$ and $\beta$ are linear maps.

(3.1.3) The inversion $~\iota:V\longrightarrow V,\
x\longmapsto \iota(x):=x^{-1}$ is a linear map and the following condition is verified:\
$(1)~~~~~~~~~~~ x+x^{-1}=\alpha(x)+\beta(x),\ \mbox{for all } x\in
V.$

(3.1.4) The map $~m:V_{(2)}:=\{(x,y)\in V\times V~|~\alpha(y)=\beta(x)\}
