Remarks on splittings in the variety of residuated lattices (Q2752413)

A residuated lattice is a bounded lattice on a set \(A\) with two additional operations, \(\cdot\) and \(\to\) such that \((A,\cdot,1)\) is a commutative monoid and the condition \(x\cdot y\leq z\) iff \(x\leq y\to z\) is satisfied. As might be expected, with so much structure, the variety \(\mathcal R\) of residuated lattices is extremely well-behaved. The aim of this paper is to show that the two-element Boolean algebra is the only algebra which splits the lattice of subvarieties of \(\mathcal R\), where a splitting algebra generates a subvariety \({\mathcal V}_1\) of \(\mathcal R\), which has a mate \({\mathcal V}_2\) such that \({\mathcal V}_1\) is not contained in \({\mathcal V}_2\), but every subvariety of \(\mathcal R\) either contains \({\mathcal V}_1\) or is contained in \({\mathcal V}_2\).