$B$-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.Comment: Journal version in Logical Methods in Computer Science. arXiv admin note: substantial text overlap with arXiv:1703.10938

@article{abs-1901-11010,
  title = {On properties of $B$-terms},
  author = {Mirai Ikebuchi and Keisuke Nakano},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 16, Issue 2},
  year = {2020},
  doi = {10.23638/lmcs-16(2:8)2020}
}