We show that if A is a stable basis algebra satisfying the distributivity condition, then B is a reduct of an independence algebra A having the same rank. If this rank is finite, then the endomorphism monoid of B is a left order in the endomorphism monoid of A.

This paper establishes a duality between the category of polytopes (finitely generated real convex sets considered as barycentric algebras) and a certain category of intersections of hypercubes, considered as barycentric algebras with additional constant operations.

If $A$ is a stable basis algebra of rank $n$, then the set $S_{n-1}$ of endomorphisms of rank at most $n-1$ is a subsemigroup of the endomorphism monoid of $A$. This paper gives a number of necessary and sufficient conditions for $S_{n-1}$ to be generated by idempotents. These conditions are satisfied by finitely generated free modules over Euclidean domains and by free left $T$-sets of finite rank, where $T$ is cancellative monoid in which every finitely generated left ideal is principal.

We consider algebras for which the operation PC of pure closure of subsets satisfies the exchange property. Subsets that are independent with respect to PC are directly independent. We investigate algebras in which PC satisfies the exchange property and which are relatively free on a directly independent generating subset. Examples of such algebras include independence algebras and dinitely generated free modules over principal ideal domains.

We prove that every finite distributive lattice can be represented as the congruence lattice of a finite (planar) semimodular lattice.

Let A be a finite set with |A| > 2. We describe all clones on A containing the set SA of all permutations of A among its unary operations. (A clone on A is a composition closed set of finitary operations on A containing all projections). With a few exceptions such a clone C is either essentially unary or cellular i.e. there exists a monoid M of self-maps of A containing SA such that either C = (= all essentially unary operations agreeing with some f ∊ M) or C = ∪ Гh where 1 < h ≤ |A| and Гh consists of all finitary operations on A taking at most h values. The exceptions are subclones of Burle's clone or of its variant (provided |A| is even).