  In this paper we introduce the notion of the center $ZBin(X)$ in the
semigroup $Bin(X)$ of all binary systems on a set $X$, and show that if
$(X,\bullet)\in ZBin(X)$, then $x\not=y$ implies $\{x,y\}=\{x\bullet y,y\bullet
x\}$.Moreover, we show that a groupoid $(X,\bullet )\in ZBin(X)$ if and only if
it is a locally-zero groupoid.
