$n$-Ary generalized Lie-type color algebras admitting a quasi-multiplicative basis

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 $\mathfrak{L}$ 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$\mathbb W$isquasi-multiplicativeifgiven$0 < k < n,$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}}$forsome$j_{\sigma} \in I;$ the product of elements of the basis $⟨e_{i_1},\dots ,e_{i_n}⟩$ belongs to $\mathbb{F}{e}_j$ for some $j\in I$ or to $\mathbb{V}$, and a similar condition is verified for the product $⟨\mathbb{V},\dots ,\mathbb{V}⟩$. We state that if $\mathfrak{L}$ admits a quasi-multiplicative basis then it decomposes as $\mathfrak{L}=\mathcal{U}\oplus (\sum {\mathfrak{J}}_k)$ with any ${\mathfrak{J}}_k$ a well described color gLt-ideal of $\mathfrak{L}$ admitting also a quasi-multiplicative basis, and $\mathcal{U}$ a linear subspace of $\mathbb V$. Also the minimality of $\mathfrak{L}$ 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 $\mathfrak{L}$.