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[VASILE POPUŢA and GEORGHE IVAN]{}\

[ABSTRACT. The main purpose of this paper is to study the vector groupoids. This is an algebraic structure which combines the concepts of Brandt groupoid and vector space such that these are compatible. The new concept of vector groupoid has applications in geometry and other areas.]{} [[^1]]{}

INTRODUCTION
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A groupoid, also known as a [*virtual group*]{} [@rams71], is an algebraic structure introduced by H. Brandt [@brandt]. A groupoid (in the sense of Brandt) can be thought as a set with a partially defined multiplication, for which the usual properties of a group hold whenever they make sense.

A generalization of Brandt groupoid has appeared in [@ehre50]. C. Ehresmann added further structures ( topological and differentiable as well as algebraic) to groupoids.

Groupoids and its generalizations (topological groupoids, Lie groupoids, measure groupoids, sympectic groupoids etc.) are mathematical structures that have proved to be useful in many areas of science \[algebraic topology ([@brow88], [@dumoiv]), harmonic analysis and operators algebras ([@dumoiv], [@renaul], [@west]), differential geometry and its applications ([@canwei], [@cdw87], [@mackey63], [@ramsren01], [@weigro]), noncommutative geometry ([@cone94]), algebraic and geometric combinatorics ([@john], [@ziv06]), dynamics of networks ([@diaste04], [@golust06], [@stegopi03] and more\].

It is remarkable to note that 