  We continue the investigation of Boolean-like algebras of dimension n (nBA)
having n constants e1,...,en, and an (n+1)-ary operation q (a "generalised
if-then-else") that induces a decomposition of the algebra into n factors
through the so-called n-central elements. Varieties of nBAs share many
remarkable properties with the variety of Boolean algebras and with primal
varieties. Exploiting the concept of central element, we extend the notion of
Boolean power to that of semiring power and we prove two representation
theorems: (i) Any pure nBA is isomorphic to the algebra of n-central elements
of a Boolean vector space; (ii) Any member of a variety of nBAs with one
generator is isomorphic to a Boolean power of this generator. This gives a new
proof of Foster's theorem on primal varieties.
