  We prove a characterization of profinite algebras, i.e., topological algebras
that are isomorphic to a projective limit of finite discrete algebras. In
general profiniteness concerns both the topological and algebraic
characteristics of a topological algebra, whereas for topological groups,
rings, semigroups, and distributive lattices, profiniteness turns out to be a
purely topological property as it is is equivalent to the underlying
topological space being a Stone space.
  Condensing the core idea of those classical results, we introduce the concept
of affine boundedness for an arbitrary universal algebra and show that for an
affinely bounded topological algebra over a compact signature profiniteness is
equivalent to the underlying topological space being a Stone space. Since
groups, semigroups, rings, and distributive lattices are indeed affinely
bounded algebras over finite signatures, all these known cases arise as special
instances of our result. Furthermore, we present some additional applications
concerning topological semirings and their modules, as well as distributive
associative algebras. We also deduce that any affinely bounded simple compact
algebra over a compact signature is either connected or finite. Towards proving
the main result, we also establish that any topological algebra is profinite if
and only if its underlying space is a Stone space and its translation monoid is
equicontinuous.
