groupoid ${\cal TVG}(V,W) $**]{}. Let $W$ be a vector subspace of a vector space $V$ over $K.$ The set $~{\cal
V}:= W\times V \times W ~$ has a natural structure of vector space. The set $ {\cal V}_{0}: = \{ (w, 0, w)\in {\cal V} ~|~ w\in
W\} $ is a vector subspace of ${\cal V}$ (here $ 0$ is the null vector of $V$ ). We introduce on $ {\cal V}:= W\times V \times W $ the structure functions $ \alpha_{\cal V}, \beta_{\cal V},
\odot_{\cal V} $ and $ \iota_{\cal V}$ as follows.

For all $~(w_{1},v,w_{2})\in {\cal V} $, the source and target $
\alpha_{\cal V}, \beta_{\cal V}: {\cal V} \to {\cal V}_{0} $ are defined by

$\alpha_{\cal V}(w_{1},v, w_{2}):= (w_{1}, 0, w_{1});~~~
\beta_{\cal V}(w_{1},v, w_{2}):= (w_{2}, 0, w_{2}).$

The partially multiplication $\odot_{\cal V}: {\cal V}_{(2)} \to {\cal V} $, where

$ {\cal V}_{(2)} = \{ ( (w_{1},
v_{1}, w_{2}), (w_{2}^{\prime}, v_{2}, w_{3}) ) \in {\cal V}\times
{\cal V} ~|~ w_{2} = w_{2}^{\prime} \}$ and the inversion map $
\iota_{\cal V}: {\cal V}\to {\cal V}$ are given by

$ (w_{1}, v_{1}, w_{2})\odot_{\cal V} (w_{2}, v_{2}, w_{3}):= (
w_{1}, v_{1}+ v_{2}, w_{3});~~ \iota_{\cal V}(w_{1}, v, w_{2}):=(
w_{2}, -v, w_{1} )$.\
