Corollary

Let M, N be two lattices of V with respect to A such that N is reflexive. In order that M ⊂ N, it is necessary and sufficient that, for all p ∈ P, M_p ⊂ N_p.

The condition is obviously necessary and, if it is fulfilled, then

As M ⊂ M** = \bigcap_p M_p, certainly M ⊂ N.

Examples (1) Everyfree lattice is reflexive. (2) Take V = K. For a fractional ideal a of K to be a reflexive lattice, it is necessary and sufficient that it be a divisorial ideal by virtue of criterion (b) of Theorem 2 and § 1, no. 4, Propositions 5 and 7. (3) Let M be a lattice with respect to A; if S is a multiplicative subset of A not containing 0, Proposition 5 of no. 1 shows that S^{-1}(M*) = (S^{-1}M)*; if M is reflexive, S^{-1}M is therefore a reflexive lattice with respect to S^{-1}A.