Quasi-shuffle algebra

In the following, we assume that \(k\) is a field containing \(\mathbb{Q}\), and \(A\) is a countable set to which we refer as the set of letters. Let \(k A\) be the \(k\)-vector space generated by \(A\) and let \(\diamond\) be a \(k\)-bilinear, associative  and commutative  product on \(k A\). A monic monomial in \(k\langle A \rangle\) will be called a word.

Definition
Let \(\diamond\) be a product on \(kA\) as above. Then we define the quasi-shuffle product \(\ast_\diamond\) on \(k\langle A \rangle\) as the \(k\)-bilinear product, which satisfies \(1 \ast_\diamond w = w \ast_\diamond 1 = w\) for any word \(w\in k\langle A \rangle\) and \begin{align}\label{eq:qshdef} a w \ast_\diamond b v = a (w \ast_\diamond b v) + b (a w \ast_\diamond v) + (a \diamond b) (w \ast_\diamond v) \end{align} for any letters \(a,b \in A\) and words \(w, v \in k\langle A \rangle\).

Theorem
\( (k\langle A \rangle, \ast_\diamond)\) is a commutative \(\mathbb{Q}\)-algebra.

Examples
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