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A variant of a BCI-algebra. One can define it by taking some of the axioms for a BCI-algebra and some of the important properties of a BCI-algebra. Specifically, a BCH-algebra is a non-empty set $X$ with a constant $0$ and a binary operation $*$ satisfying the following axioms:

1) $x * x = 0$;

2) if $x * y = 0$ and $y * x = 0$, then $x = y$;

3) $(x*y)*z = (x*z)*y$.

Clearly a BCI-algebra is a BCH-algebra; however, the converse is not true. While some work has been done on such algebras, generally they have not been as extensively investigated as BCI-algebras.


[a1] Qing-ping Hu, Xin Li, "On BCH-algebras" Math. Seminar Notes (Kobe University) , 11 (1983) pp. 313–320 Zbl 0579.03047
[a2] Y. Imai, K. Iséki, "On axiom systems of propositional calculi, XIV" Proc. Japan Acad. Ser. A Math. Sci. , 42 (1966) pp. 19–22 DOI 10.3792/pja/1195522169 MR0195704 Zbl 0156.24812
[a3] K. Iséki, "An algebra related with a propositional calculus" Proc. Japan Acad. Ser. A, Math. Sci. , 42 (1966) pp. 26–29 DOI 10.3792/pja/1195522171 MR0202571 Zbl 0207.29304
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BCH-algebra. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by C.S. Hoo (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article