The authors associate with a finite monoid \(S_0\) and \(m\) finite commutative monoids \(S_1,\dots,S_m\) a product \(\lozenge_m(S_m,\dots,S_1,S_0)\). There is a representation of the free objects in the pseudovariety \(\lozenge_m({\mathcal W}_m,\dots,{\mathcal W}_1,{\mathcal W}_0)\) generated by these \((m+1)\)-ary products where \(S_i\in{\mathcal W}_i\) for all \(0\leq i\leq m\). In particular, there is given a criterion to determine when an identity holds in \(\lozenge_m({\mathcal J}_1,\dots,{\mathcal J}_1,{\mathcal J}_1)\) with the help of a version of the Ehrenfeucht-Fraïsse game, where \({\mathcal J}_1\) denotes the pseudovariety of all semilattice monoids. The union \(\bigcup_{m>0}\lozenge_m({\mathcal J}_1,\dots,{\mathcal J}_1,{\mathcal J}_1)\) turns out to be the second level of Straubing's dot-depth hierarchy of aperiodic monoids.