Combining Proposition 4.7 and Proposition 4.9, we can get a characterization of a modular lattice by the fixed set of a derivation.
Namely, in a vector space the subspaces (i.e. the subsets closed under linear dependence) form a modular lattice; but the algebraically closed subsets of an algebraically closed field form a lattice that is not modular.
In order to get information about the free modular lattice in n generators, they described part of the lattice [ → Q ( a ) 〉 where Δ is the n -subspace quiver and Q ( a ) is the indecomposable injective k Δ -module with a the sink of Δ. Of course, for n ≥ 4, this lattice [ → Q ( a ) 〉 is of infinite height!
In the following, we give a property of principal ideals in a modular lattice.
Two of such alternatives are the modular lattice architectures proposed by Lev-ari[26], and Glentis and Kalouptsidis[27].
Using the fixed sets of isotone derivations, we establish characterizations of a chain, a distributive lattice, a modular lattice and a relatively pseudo-complemented lattice, respectively.
Proof Let L be a modular lattice and I = 〈 a 〉 be a principal ideal of L. Assume x, y ∈ L and x ∼ y.
Now, using fixed sets of derivations, we give a condition by which a lattice becomes a modular lattice.
Note that the picture is obtained from the free modular lattice in 3 generators as presented by Dedekind [11] in 1900 by inserting in the non-distributive interval of length 2 further diagonals (one may call it the free k -modular lattice in 3 generators).
