The study of the recently introduced notions of amenability, congeniality and simplicity of bases for infinite dimensional algebras is furthered. A basis B over an infinite dimensional F-algebra A is called amenable if F B, the direct product indexed by B of copies of the field F, can bemadeinto an A-module in a natural way. (Mutual) congeniality is a relation that serves to identify cases when different amenable bases yield isomorphic A-modules. If B is congenial to C but C is not congenial to B, then we say that B is properly congenial to C. An amenable basis B is called simple if it is not properly congenial to any other amenable basis and it is called projective if there does not exist any amenable basis which is properly congenial to B. We introduce a family of algebras and study these notions in that context; in particular, we show that the family includes examples of algebras without simple or projective bases. This same family also serves to illustrate the one-sided nature of amenability and simplicity as we produce examples of bases which are amenable only on one side and, likewise, bases which are only one one-sided simple.