  A new 3-ary non-associative algebra, which is called a semi-associative
$3$-algebra, is introduced, and the double modules and double extensions by
cocycles are provided. Every semi-associative $3$-algebra $(A, \{ , , \})$ has
an adjacent 3-Lie algebra $(A, [ , , ]_c)$. From a semi-associative $3$-algebra
$(A, \{, , \})$, a double module $(\phi, \psi, M)$ and a cocycle $\theta$, a
semi-direct product semi-associative $3$-algebra $A\ltimes_{\phi\psi} M $ and a
double extension $(A\dot+A^*, \{ , , \}_{\theta})$ are constructed, and
structures are studied.
