)$. Therefore, each vector space $
V $ over $K$ can be regarded as vector groupoid over $V_{0}
=\{0\}$.$\b$

Let $V$ be a vector space over a field $K$. Then $ V$ has a structure of null groupoid over $V$ ( see Example 2.1(ii) ). In this case the structure functions are $~\alpha = \beta =\iota =
Id_{V}$ and $ x \odot x = x $ for all $x\in V$. We have that $V_{0} = V$ and the maps $ \alpha, \beta, \iota $ are linear. Since $ x+\iota(x) = x+ x $ and $ \alpha(x) + \beta(x) = x+x $ imply that the condition 3.1.3(1) holds. It is easy to verify the conditions 3.1.4(1)- 3.1.4(4) from Definition 3.1. Then $V$ is a vector groupoid, called the [*null vector groupoid*]{} associated to $V$.$\b$

Let $V$ be a vector space over a field $K$. We consider the pair groupoid $( V\times V, {\widetilde}{\alpha}, {\widetilde}{\beta},
{\widetilde}{m},{\widetilde}{\iota}, \Delta_{V})$ associated to $V$ ( see Example 2.1(iii)). We have that $ V\times V$ is a vector space over $K$ and the source ${\widetilde}{\alpha}$ and target ${\widetilde}{\beta}$ are linear maps. Also, the inversion ${\widetilde}{\iota} : V\times V \to V\times V$ is a linear isomorphism. Therefore it 