A partition of the positive integers into sets A and B avoids a set S ⊂ N if no two distinct elements in the same part have a sum in S. If the partition is unique, S is uniquely avoidable. For any irrational α > 1, Chow and Long constructed a partition which avoids the numerators of all convergents to α, and conjectured that the set Sα which this partition avoided was uniquely avoidable. We pro...

An algebra A is uniquely generated by a set G if G is a subset of every set that generates A. We investigate uniquely generated algebras and focus especially on equational classes of algebras in which the free algebras are uniquely generated. We show that such classes possess a number of algebraic properties that are in some sense extremal. We also present algebraic conditions on an equational ...

It is well known that every uniquely clean ring is strongly clean. In this paper, we investigate the question of when this result holds element-wise. We first construct an example showing that uniquely clean elements need not be strongly clean. However, in case every corner ring is clean the uniquely clean elements are strongly clean. Further, we classify the set of uniquely clean elements for ...