according to A. Connes [@cone94], Heisenberg was discovered quantum mechanics by considering the groupoid of quantum transitions rather than the group of symmetry.

The paper is organized as follows. In Section 2 we define groupoids and useful properties of them are presented. In Section 3 we introduce the concept of vector groupoid and its properties are established. In Section 3 we give some algebraic constructions of vector groupoids.

BRANDT GROUPOIDS
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We recall the minimal necessary backgrounds on groupoids for our developments (for further details see e.g. [@brow87], [@higg71], [@ivan02], [@vpop] and references therein for more details).

([@cdw87]) A **groupoid $ G $ over** $ G_{0} $ ( **in the sense of Brandt** ) is a pair $
(G, G _0) $ of nonempty sets such that $ G _0\subseteq G  $ endowed with two surjective maps $\alpha ,\beta :G \rightarrow G
_0$ ( called [**source**]{}, respectively [**target**]{}, a partially binary operation (called [**multiplication**]{}) $~m :G
_{(2)}\rightarrow G,~(x,y)\longmapsto m \left( x,y\right) :=x\cdot
y,~$ where $~G _{(2)}:= G\times_{(\beta, \alpha)}G = \left\{
\left( x,y\right) \in G \times G \mid \beta \left( x\right)
=\alpha \left( y\right) \right\}$ is the [**set of composable pairs**]{} and a map $~\iota :G \rightarrow G,~x\longmapsto \iota
(x):=x^{-1}$ ( 