CONJECTURES OF ENE, HERZOG, HIBI, AND SAEEDI MADANI IN THE JOURNAL OF ALGEBRA

In their 2014 preprint, “pseudo-gorenstein and level hibi rings,” ene, herzog, hibi, and saeedi madani assert (theorem 4.3) that for a regular planar lattice l with poset of joinirreducibles p, the following are equivalent: (1) l is level; (2) for all x, y ∈ p such that y l x, heightpˆ(x) + depthpˆ(y) ≤ rank(pˆ) + 1; (3) for all x, y ∈ p such that ylx, either depth(y) = depth(x)+1 or height(x) = height(y) + 1, where pˆ is the poset p with a new top and bottom adjoined. they added, “computational evidence leads us to conjecture that the equivalent conditions given in theorem 4.3 do hold for any planar lattice (without any regularity assumption).” in their 2015 journal of algebra article, ene et al. prove the equivalence of the last two conditions for a regular simple planar lattice (proposition 4.3), and write, “one may wonder whether the regularity condition in proposition 4.3 is really needed.” in this note, an example is given showing that the regularity condition cannot be dropped. in their 2015 article, ene et al. say that “we expect” the second condition to imply the first for any finite distributive lattice l. in this note, we provide a counter-example.