as unity, is a groupoid over $ G_{0} = \{ e \}$ with the structure functions $ \alpha,
\beta, m, \iota $ given by:\
$\alpha(x) = \beta(x) = e,~ \iota(x)
= x^{-1}$ for all $ x\in G$ and $ m(x,y)= xy $ for all $x,y\in G$.

[*(ii)*]{}  Any set $~X~$ can be endowed with a [*nul groupoid*]{} structure over itself. For this we take: $~\alpha = \beta =\iota =
Id_{X}; ~x , y\in X~$ are composable iff $~x=y~$ and we define $~
x\cdot x = x.$

[*(iii)*]{}  The Cartesian product $~G:= X \times X~$ has a structure of groupoid over $ \Delta_{X} = \{ (x,x)\in X\times X
~|~ x\in X \}$ by taking the structure functions as follows: $~{\widetilde}{{\alpha}}(x,y):= (x,x),~ {\widetilde}{{\beta}}(x,y):= (y,y);~$ the elements $~ (x,y)~$ and $~(y^{\prime},z)~$ are composable in $~G:=X \times X~$ iff $~y^{\prime} = y~$ and we define $~(x,y)\cdot (y,z) = (x,z)~$ and the inverse of $~(x,y)~$ is defined by $~(x,y)^{-1}:= (y,x).~$ This is usually called the [*pair*]{} or [*coarse groupoid*]{}. Its unit set is $
G_{0}:=\Delta_{X}.~$ The isotropy group $~G(u)~$ at $~u=(x,x)~$ is the nul group $~\{(u,u)\}.$

[*(i)*]{} [*The symmetry groupoid $ {\cal
SG}(X) $*]{}. Let $ X $ be a nonempty set and consider\
$G:= 