Let $*$ be a binary operation on $X$. We define the following properties of $*$:
Property Condition $A$ is closed under $*$ \(x*y\in A\) Commutative \(x*y=y*x\) Associative \(x*(y*z)=(x*y)*z\) Idempotent \(x*x=x\) Left-cancellative \(x*y=x*z\to y=z\) Right-cancellative \(y*x=z*x\to y=z\) Cancellative Left- and Right-cancellative Central \((x*y)*(y*z)=y\) If $+$ is another binary operation on $X$, we further define:
Property Condition Left-distributive over $+$ \(x*(y+z)=x*y+x*z\) Right-distributive over $+$ \((y+z)*x=y*x+z*x\) Distributive over $+$ Left- and Right-distributive over $+$ The statements in the Condition column are meant to hold either for all $x\in X$; all $x$, $y\in X$; or all $x$, $y$, $z\in X$, accordingly.