On the Computational Power of Programs over $$\mathsf {BA}_2$$ Monoid

The PLP conjecture for monoids states that for every monoid M, either M is universal (that is, for every language \(L \subseteq \varSigma ^*\) there is a program over M which accepts the language L) or it has the polynomial length property (that is, every program over the monoid M has an equivalent program of length \({\mathsf {poly}}(n)\)). The conjecture has been confirmed (Tesson-Therien (2001)) for the case of groups and several subclasses of aperiodic monoids such as the variety DA and the monoids divided by the monoid U. However, the case of the set of monoids divided by the monoid \(\mathsf {BA}_2\) is still open, which if resolved, confirms the conjecture for all aperiodic monoids.