Definition 1. 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.