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 $\struct {S, \circ}$ is right cancellable if and only if:

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

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

Cancellable Operation

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

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

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