On properties of $B$-terms

$B$-terms are built from the $B$ combinator alone defined by $B\equiv\lambda f.\lambda g.\lambda x. f~(g~x)$, 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 property and show that there are infinitely many $B$-terms which do not have the property. Also, we introduce a canonical representation of $B$-terms that is useful to detect cycles, or equivalently, to prove the property, with an efficient algorithm.