  We study the quadratic algebras $A(K,X,r)$ associated to a class of strictly
braided but idempotent set-theoretic solutions $(X,r)$ of the Yang-Baxter or
braid relations. In the invertible case, these algebras would be analogues of
braided-symmetric algebras or `quantum affine spaces' but due to $r$ being
idempotent they have very different properties. We show that all $A(K,X,r)$ for
$r$ of a certain permutation idempotent type are isomorphic for a given
$n=|X|$, leading to canonical algebras $A(K,n)$. We study the properties of
these both via Veronese subalgebras and Segre products and in terms of
noncommutative differential geometry. We also obtain new results on general PBW
algebras which we apply in the permutation idempotent case.
