Every finite lattice in \(\mathcal V(\mathbf M_ 3)\) is representable (Q2496160)

A finite lattice is representable if it is isomorphic to the congruence lattice of a finite algebra. It is well-known that every finite distributive lattice is representable (R.\, W.\, Quackenbush, 1971) and that every finite lattice in the variety generated by \({\mathbf N}_{5}\) is representable. The author proves that also every finite lattice in a variety generated by \({\mathbf M}_{3}\) is representable (although \({\mathbf M}_{3}\) is not fermentable contrary to \({\mathbf N}_{5}\) or the two-element lattice).