Residually small varieties without rank (Q702200)

We recall that a variety \(\mathbb V\) is residually small if any class of non-isomorphic subdirectly irreducible algebras from \(\mathbb V\) is a set. For a set functor \(F\), let \(\text{COAL}(F)\) denote the category of all \(F\)-coalgebras and their homomorphisms and let \(\mathbb V_F\) be a variety of (unranked) algebras equivalent to the dual of \(\text{COAL}(F)\). It is proved that the variety of complete atomic Boolean algebras and the variety of commutative \(C^{*}\)-unital algebras are residually small and Birkhoff's subdirect representation theorem is true for these varieties. On the other hand, the variety of compact Hausdorff spaces is residually small but Birkhoff's subdirect decomposition theorem does not hold. These observations are generalized. If \(F\) is a set-accessible functor preserving intersections then Birkhoff's subdirect decomposition theorem holds in \(\mathbb V_F\) and \(\mathbb V_F\) is residually small.