| Summary: | $B$-terms are built from the $B$ combinator alone defined by $B\equiv\lambda
fgx. f(g~x)$, 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.
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