  The class of generalized Lie-type color algebras contains the ones of
generalized Lie-type algebras, of $n$-Lie algebras and superalgebras,
commutative Leibniz $n$-ary algebras and superalgebras, among others. We focus
on the class of generalized Lie-type color algebras $\frak L$ admitting a
quasi-multiplicative basis, with restrictions neither on the dimensions nor on
the base field $\mathbb F$ and study its structure. If we write $\frak L =
\mathbb V \oplus \mathbb W$ with $\mathbb V$ and $0 \neq \mathbb W$ linear
subspaces, we say that a basis of homogeneous elements $\mathfrak{B} =
\{e_i\}_{i \in I}$ of $\mathbb W$ is quasi-multiplicative if given $0 < k < n,$
for $i_1,\dots,i_k \in I$ and $\sigma \in \mathbb S_n$ satisfies $\langle
e_{i_1}, \dots, e_{i_k}, \mathbb V, \dots, \mathbb V \rangle_{\sigma} \subset
\mathbb{F}e_{j_{\sigma}}$ for some $j_{\sigma} \in I;$ the product of elements
of the basis $\langle e_{i_1}, \dots, e_{i_n} \rangle$ belongs to
$\mathbb{F}e_j$ for some $j \in I$ or to $\mathbb V$, and a similar condition
is verified for the product $\langle \mathbb V, \dots,\mathbb V \rangle$. We
state that if $\frak L$ admits a quasi-multiplicative basis then it decomposes
as $\mathfrak{L} ={\mathcal U} \oplus (\sum\limits {\frak J}_{k})$ with any
${\frak J}_k$ a well described color gLt-ideal of $\frak L$ admitting also a
quasi-multiplicative basis, and ${\mathcal U}$ a linear subspace of $\mathbb
V$.
  Also the minimality of $\frak L$ is characterized in terms of the connections
and it is shown that the above direct sum is by means of the family of its
minimal color gLt-ideals, admitting each one a $\mu$-quasi-multiplicative basis
inherited by the one of $\frak L$.
