View source for BCH-algebra

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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 relation|binary relation]] $*$ 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.

====References====
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  Qing-ping Hu,   Xin Li,   "On BCH-algebras"  ''Math. Seminar Notes (Kobe University)'' , '''11'''  (1983)  pp. 313–320 {{ZBL|0579.03047}}</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  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}} {{MR|0195704}} {{ZBL|0156.24812}}</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  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}} {{MR|0202571}} {{ZBL|0207.29304}}</TD></TR>
</table>