It is known that a lattice $L$ is distributive if and only if it does not contain the diamond $M_3$ or the pentagon $N_5$ as sublattices.
A complete lattice is one in which every subset has an infimum and a supremum.
A frame is a complete lattice $L$ satisfying the infinite distributive law $x\wedge(\bigvee_iy_i)=\bigvee_i(x\wedge y_i)$ for all $x,\{y_i\}_i\in L$. This is equivalent to $L$ being a complete Heyting algebra, that is, a complete lattice where the $x\wedge-$ map has a right adjoint.
As frames are lattices satisfying a stronger form of distributivity than distributive lattices, it seems possible that they could also be characterized by the nonexistence of embeddings from a certain family of "minimal" non-frame lattices. Intuitively I imagine these lattices would in some way generalize $M_3$ and/or $N_5$, but include infinite ascending chains of some sort. However, I am struggling to formulate a precise characterization.
I would not be surprised if our desired family has proper class many elements, to accommodate for the different cardinalities the infinite chain may have.
Furthermore, the proofs of the $M_3,N_5$ theorem I have seen involve the use of free lattices. But free complete lattices do not in general exist, due to how big they would have to be (see Johnstone's Stone Spaces, I.4.7). So I am unsure of how to go about generalizing the theorem. I have found nothing in the literature regarding this question.
Can frames, or other classes of distributive lattices such as completely distributive lattices, be characterized by the nonexistence of certain types of sublattices? Thank you!
