Let $*$ be a binary operation on a set $X$. 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.