A preframe is a poset with finite meets and directed joins, and in which the finite meets distribute over the directed joins.
A preframe is a frame if and only if it is also a distributive lattice, i.e. it has finite joins which are distributed over by finite meets.
Preframes are a useful technical tool in the study of proper maps of locales.
Since preframes involve operations of arbitrarily large arity, it is not automatic that they are monadic functor over Set or that they can be presented by generators and relations. It is, however, true; see