Cancellation

https://en.wikipedia.org/wiki/Cancellation_property

The notion of cancellative is a generalization of the notion of invertible.

1. An element a in a magma (M, ∗) has the left cancellation property (or is left-cancellative) if for all b and c in M, a ∗ b = a ∗ c always implies that b = c.

a ∈ Magma (M, ∗) is left-cancellative if ∀bc ∈ M. (a ∗ b = a ∗ c) -> b = c

1. An element a in a magma (M, ∗) has the right cancellation property (or is right-cancellative) if for all b and c in M, b ∗ a = c ∗ a always implies that b = c.

a ∈ Magma (M, ∗) is right-cancellative if ∀bc ∈ M. (b ∗ a = c ∗ a) -> b = c

1. An element a in a magma (M, ∗) has the two-sided cancellation property (or is cancellative) if it is both left- and right-cancellative.

a ∈ Magma (M, ∗) is cancellative if ∀bc ∈ M. (a∗b=a∗c) ∨ (b∗a = c∗a) -> b = c

1. A magma (M, ∗) has the left cancellation property (or is left-cancellative) if all a in the magma are left cancellative.

2. A magma (M, ∗) has the right cancellation property (or is right-cancellative) if all a in the magma are right cancellative.

3. A magma (M, ∗) has the cancellation property (or is cancellative) if all a in the magma are cancellative.

4. A left-invertible element is left-cancellative.

5. A right-invertible element is right-cancellative.

6. A invertible element is cancellative.

For example, every quasigroup, and thus every group, is cancellative.