  To describe external and internal attributes of fundamental fermions, a
theory of multi-spinor fields is developed on an algebra, a {\it triplet
algebra}, which consists of all the triple-direct-products of Dirac
\gamma-matrices. The triplet algebra is decomposed into the product of two
subalgebras, an external algebra and an internal algebra, which are exclusively
related with external and internal characteristic of the multi-spinor field
named {\it triplet fields}. All elements of the external algebra which is
isomorphic to the original Dirac algebra $A_\gamma$ are invariant under the
action of permutation group $S_3$ which works to exchange the order of the
$A_\gamma$ elements in the triple-direct-product. The internal algebra is
decomposed into the product of two $4^2$ dimensional algebras, called the
family and color algebras, which describe the family and color degrees of
freedom. The family and color algebras have fine substructures with "trio plus
solo" conformations which are irreducible under the action of $S_3$. The
triplet field has trio plus solo family modes with ordinary tricolor quark and
colorless solo lepton components. To incorporate the Weinberg-Salam mechanism,
it is required to introduce two types of triplet fields, a left-handed doublet
and right-handed singlets of electroweak iso-spin. It is possible to qualify
the Yukawa interaction and to make a new interpretation of its coupling
constants naturally in an intrinsic mechanism of the triplet field formalism.
The ordinary Higgs mechanism leads to a new type of the Dirac mass matrices
which can explain all data of quark sector within experimental accuracy.
