Pointfree topology: can frames be characterized in terms of forbidden substructures?

Question

Does there exist a class of complete lattices $\mathcal{C}$ such that for all complete lattice $L$, the following are equivalent?

1. In $L$, finite meets distribute over arbitrary joins. i.e. $L$ is a frame.
2. There is no way of embedding an element of $\mathcal{C}$ into $L$ such a way as to preserve arbitrary non-empty meets and joins.

In other words, can frames be characterized in terms of forbidden substructures?

Answer 1

Just to embellish on user138171's answer, the key point of that argument is that the class of frames is closed under substructure. Exactly the same argument shows that a class of structures can be characterized in terms of forbidden substructures if and only if it is closed under substructure.

Further, a class of structures is closed under substructure if and only if it can be axiomatized by a (possibly infinitary) universal theory, i.e. by conditions of the form $\forall \overline{x} \phi(\overline{x})$, where $\overline{x}$ is a possibly infinite tuple and $\phi$ is infinitary quantifier-free. I bring this up because the defining condition of frames is obviously a universal condition.

To prove the claim in the last paragraph, observe that (infinitary) universal conditions are obviously closed under substructure, and conversely, the class $K$ can be axiomatized by the following axioms, one for every structure $A$ not in $K$: $\forall \overline{x}\,\lnot \phi_A(\overline{x})$, where $\phi_A(\overline{x})$ is the diagram of $A$, i.e. asserts that the variables $\overline{x}$ label a structure isomorphic to $A$.

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Edit: I'm working in the language with symbols for infinitary meet and join (actually, for frames, I believe join will do, but goblin said he wanted to characterize frames among complete lattices). This is equivalent to user138171's assertion that "embedding" means "embedding of complete lattices", i.e. homomorphisms preserve infinitary meet and join.

Of course, there are some technical concerns here. If these are to be symbols in the usual sense, they should have arities, so maybe we need an infinitary join symbol of arity each infinite cardinal... but then we might worry that we have a proper class of symbols... These issues make "infinitary universal algebra" not quite as nice as usual universal algebra - some free algebras may not exist, for example - but they don't matter for the simple argument above.

It's easy to give an (infinitary) equational axiomatization of complete lattices in this language - just take the usual equational axiomatization of lattices and add generalized infinitary versions of associativity. I'll leave it to you to check that this, together with the usual idempotence, implies that the infinitary join and meets act the way they're supposed to.

Answer 2

Let $\mathcal C$ be the class of all lattices which are not embedded in any frame. Suppose none of $\mathcal C$ embed in a lattice $L$. If $L$ is not a frame, clearly it cannot be embedded in any frame. So $L\in \mathcal C$ and it is embedded in $L$. A contradiction. Therefore $L$ must be a frame.

As noted above by lattice or embedding I mean a complete one.