called **inversion**), which verify the following conditions:

(G) ([**associativity**]{}): $~(x\cdot y)\cdot z=x\cdot (y\cdot z)$ in the sense that if one of two products $(x\cdot y)\cdot z$ and $x\cdot (y\cdot z)$ is defined, then the other product is also defined and they are equals;

(G2) ([**units**]{}): for each $x\in G~ \Rightarrow ~(\alpha(x),x),
~(x,\beta( x))\in G _{(2)}~$ and we have $~\alpha(x)\cdot
x=x\cdot \beta(x)=x $;

(G3) ([**inverses**]{}): for each $ x\in G~\Rightarrow ~ (x,x^{-1}), ~(x^{-1},x)\in G
_{(2)} $ and we have $~x^{-1}\cdot x=\beta(x), ~~x\cdot
x^{-1}=\alpha(x).$

A groupoid $G$ over $G _0$ with the *structure functions* $
\alpha ,\beta ,m ,\iota $ is denoted by $(G, \alpha, \beta, m,
\iota, G _0)$ or $(G, \alpha, \beta, G _0)$ or $(G, G _0)$. The element $\alpha(x)$ respectively $\beta(x)$ is called the *left unit* respectively *right unit* of $x;$ $~G_0$ is called the *unit set* of $G$. The map $(\alpha,
\beta)$ defined by: $$(\alpha,\beta):G\rightarrow G _0\times G _0,\quad (\alpha,
\beta)(x):=(\alpha(x), \beta(x)),~x\in G,$$ is called the *anchor map* of $G$. For each $ u\in G_{0} $, the set $
G_{u}:= \alpha^{-1}(u)$ ( resp. $ G_{u}:= \beta^{-1}(u)$ ) is called *$\alpha-$ fibre* ( resp. *$\beta-$ fibre* ) of $ G$ at $ u\in G_{0}$. If $~u,v\in G_{0}~$ we 