The Lattice of all subvarieties of a variety

Question

Suppose that we have a variety $V$ and the corresponding lattice of all subvarieties. My question is: is that lattice an object of our ZFC theory? Especially, is it small? If we take and element of it, say $V$ itself, then $\bigcup V$is not a small set, hence the entire lattice is not small either. I do not think that this question is particularly interesting, but I would like to see how it is formally treated that elements of supposedly small thing are not small themselves.

Answer 1

A variety as usually defined is of course a class, but you can encode the same information by identifying a variety with the set of identities that hold in it. That is, you can instead define a variety over a signature to be a set of identities over that signature which is closed under deduction. This makes the collection of all varieties over a given signature into a set. (The "subvariety" order then corresponds to reverse inclusion on sets of identities: a set of identities $S$ corresponds to a subvariety of a set of identities $T$ if $S\supseteq T$.)