# Definition:Cancellable Operation

## Definition

Let $\left ({S, \circ}\right)$ be an algebraic structure.

### Left Cancellable Operation

The operation $\circ$ in $\struct {S, \circ}$ is left cancellable if and only if:

$\forall a, b, c \in S: a \circ b = a \circ c \implies b = c$

That is, if and only if all elements of $\struct {S, \circ}$ are left cancellable.

### Right Cancellable Operation

The operation $\circ$ in $\left ({S, \circ}\right)$ is right cancellable iff:

$\forall a, b, c \in S: a \circ c = b \circ c \implies a = b$

That is, iff all elements of $\left ({S, \circ}\right)$ are right cancellable.

### Cancellable Operation

The operation $\circ$ in $\left ({S, \circ}\right)$ is cancellable iff:

• $\forall a, b, c \in S: a \circ b = a \circ c \implies b = c$
• $\forall a, b, c \in S: a \circ c = b \circ c \implies a = b$

That is, iff it is both a left cancellable operation and a right cancellable operation.