# Right-cancellative

## Definition 2

Let $$(S, \circ)$$ be an algebraic structure. Then $$\circ$$ is right-cancellative if and only if

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

Natural number addition is right-cancellative.

$$\forall a,b,c \in \mathbb{N} : b + a = c + a \Rightarrow b = c$$

Proof available

LEMMA

Integer addition is right-cancellative.

$$\forall x,y,z \in \mathbb{Z} : x + z = y + z \Rightarrow x = y$$

Proof available

THEOREM

The left operation is right-cancellative.

$$y \leftarrow x = z \leftarrow x \Rightarrow y = z$$

Proof available