The authors first set up the algebraic framework (stuffle and shuffle products) of multiple zeta values (MZVs). The system of the regularized double shuffle relations is a conjectured full relation system of multiple zeta values, that is, any other algebraic relation among MZVs should follow from these. The authors take the task of deducing as many relations from these basic ones as many possible (in Section 3).
One representative relation, for instance, is \[ \sum_{j=1}^{n}(-1)^{n-j}\left(\begin{array}{c}k-j-1\\ k-n-1\end{array}\right)\sum_{\begin{array}{c}\operatorname{wt}(\mathbf{k})=k, \operatorname{dep}(\mathbf{k})=j \\ k \text { is admissible }\end{array}} \zeta_{R}^{\star}(\mathbf{k})=\zeta_{R}(k). \] Here \[ \zeta_{R}(\mathbf{k})=Z_{R}\left(z_{k_{1}} \cdots z_{k_{n}}\right), \quad \zeta_{R}^{\star}(\mathbf{k})=Z_{R}^{\star}\left(z_{k_{1}} \cdots z_{k_{n}}\right) \] are defined via mappings from the \(\mathfrak{h}^{0}\) subalgebra of non-commutative bi-variate polynomials to the scalars.
The system of the regularized double shuffle relations is, of course, not the only possible full relation system. The closing section of the paper (Section 4) reviews some other possibilities.