On properties of B-terms

B-terms are built from the B combinator alone defined by B ≡ λfgx.f(g x), which is well known as a function composition operator. This paper investigates an interesting property of B-terms, that is, whether repetitive right applications of a B-term cycles or not. We discuss conditions for B-terms to have and not to have the property through a sound and complete equational axiomatization. Specifically, we give examples of B-terms which have the cyclic property and show that there are infinitely many B-terms which do not have the property. Also, we introduce another interesting property about a canonical representation of B-terms that is useful to detect cycles, or equivalently, to prove the cyclic property, with an efficient algorithm.