Splitting in the variety of residuated lattices (Q1866846)

A pair \(( {\mathcal V}_1, {\mathcal V}_2)\) of subvarieties of a variety \(\mathcal V\) is called a splitting pair if \({\mathcal V}_1 \not \subseteq {\mathcal V}_2\) and for any subvariety \(\mathcal S\) of \(\mathcal V\) either \({\mathcal V}_1 \subseteq \mathcal S\) or \({\mathcal S} \subseteq {\mathcal V}_2\). In such a case, \({\mathcal V}_1\) is generated by an \(SI\)-algebra which is called a splitting algebra. The authors show that the only algebra that splits the lattice of subvarieties of the variety of residuated lattices is the two-element Boolean algebra.