Abstract

This paper offers an analysis of the effect of the identity combinators in dual systems. The result is based on an easy technical trick, namely, that the identity combinators collapse all the combinators which are dual with respect to them. After reviewing dual combinators I consider the possible combinatory systems and l-calculi in which the functions and/or the application operation are bidirectional. The last section of the paper shows the devastating effect the identity combinators have for a dual system: they half trivialize simple combinatory bases, although they are not sufficient to cause real triviality for what cancellative combinators are needed.