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I am having trouble understanding one argument in the proof of theorem 58.6 of "Graded Syzygies" by Irena Peeva. The theorem is

Let $L$ be a finite atomic lattice. There exists a monomial Ideal whose lcm-lattice is $L$

At the end of the proof, we look at an element $c$ of the lattice $L$ and observe the set of meet irreducible elements over it $M_c$. The proof next claims that

Note that $c$ is the meet of the elements in $M_c$

Why is that? I understand that every element $c$ in the lattice is the join of the atoms $p_1, p_2, ..., p_q$ such that $c$ is over these atoms but can't seem to make a similar argument for the claim above. Would really appreciate clarification here.

Roni Varshavsky
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  • Doesn't duality do it? Just reverse <= – Mariano Suárez-Álvarez Feb 22 '23 at 15:21
  • How would you do that? For atoms, we know that every element must be the join of atoms by assumption that $L$ is atomic, but I'm not sure if we can automatically reverse that. Moreover, meet irreducible elements might have elements between them and the maximal element (unlike atoms with the minimal element) – Roni Varshavsky Feb 22 '23 at 16:35
  • Every finite poset is both relatively atomic and relatively coatomic. Finite lattices are bounded. Thus finite lattices are both atomic and coatomic.

    So the "atomic" assumption is redundant in your theorem, and given that the lattice is coatomic, too, it seems that you already know the proof.

     – Badam Baplan Feb 23 '23 at 20:27
  • "Graded Syzygies" defines "atomic" as a lattice such that every non bottom element is the join of atoms, which I believe is not true for any lattice (take a diamond lattice and add an element on top of the maximal element to get a non atomic lattice) and therefore this assumption is not redundant. Yet I cannot automatically see why this assumption gives what I need in the other direction. – Roni Varshavsky Feb 24 '23 at 15:23

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