A BCL algebra is a universal algebra with a binary operation denoted as "$*$" and a $0$-ary operation (constant) denoted as "$0$", satisying the following axioms:
(1) $x * x = 0$;
(2) if $x * y = 0$ and $y * x = 0$, then $x=y$;
(3) $(((x * y) * z) * ((x * z) * y)) * ((z * y) * x) = 0$
This definition is from the article "A New Branch of the Pure Algebra: BCL-Algebras" by Yonghong Liu published in Advances in Pure Mathematics, 2011, 1, 297-299.
The author also defines a relationship $x\leqslant y$ iff $x*y = 0$. Without any proof, this relationship is said to be partial order. While it is obviously reflexive and antisymmetric, I find it impossible to proof that it is also transitive. Can anybody help?