  Standard model is reconstructed using the generalized differential calculus
extended on the discrete space $M_4\times Z_3$. $Z_3$ is necessary for the
inclusion of strong interaction. Our starting point is the generalized gauge
field expressed as $A(x,y)=\!\sum_{i}a^\dagger_{i}(x,y){\bf d}a_i(x,y),
(y=0,\pm)$, where $a_i(x,y)$ is the square matrix valued function defined on
$M_4\times Z_3$ and ${\bf d}=d+{\chi}$ is generalized exterior derivative. We
can construct the consistent algebra of $d_{\chi}$ with the introduction of the
symmetry breaking function $M(y)$ and the spontaneous breakdown of gauge
symmetry is coded in ${d_{\chi}}$.
  The gauge field $A_\mu(x,y)$ and Higgs field $\Phi(x,y)$ are written in terms
of $a_i(x,y)$ and $M(y)$, which might suggest $a_i(x,y)$ to be more fundamental
object. The unified picture of the gauge field and Higgs field as the
generalized connection in non-commutative geometry is realized. Not only
Yang-Mills-Higgs lagrangian but also Dirac lagrangian, invariant against the
gauge transformation, are reproduced through the inner product between the
differential forms. Two model constructions are presented, which are
distinguished in the particle assignment of Higgs field $\Phi(x,y)$.
