Algebra 6 | Cancellation Property [dark version]

Cancellation in a finite semigroup doesn’t just simplify equations—it forces the structure to be a group. The core claim is that for a finite semigroup, having both left and right cancellation properties is strong enough to guarantee the existence of an identity element and inverses for every element, turning the semigroup into a group.

The discussion starts by recalling what cancellation means in a group. If a·x = a·y, then x = y (left cancellation). If x·b = y·b, then x = y (right cancellation). These follow directly from the presence of inverses: multiplying by the inverse of the canceled element on the appropriate side collapses the equation back to x = y. The key takeaway is that in groups, cancellation is automatic.

The next step flips the direction of reasoning. It introduces a lemma that characterizes when a semigroup is actually a group without assuming an identity element at the outset. The lemma says: a semigroup is a group exactly when, for every pair of elements a and b, there exist elements x and y such that a·x = b and y·a = b. In other words, every element b can be reached from the left by multiplying a by something, and also reached from the right by multiplying something by a.

One direction of the lemma is straightforward: if inverses exist, then x can be chosen as a^{-1}·b and y as b·a^{-1}. The harder direction uses the strength of the “for all a, b” requirement. By setting b = a, the construction yields candidates that behave like an identity on the left. Then, using an arbitrary element b, the argument extends that left identity to all elements and also produces left inverses for every element. With standard semigroup facts (notably that left invertibility plus the given structure yields a group), the lemma concludes that the semigroup must be a group.

Finally, the main proposition connects cancellation properties to the lemma using finiteness. For a finite semigroup S, left cancellation means that for each fixed a, the function f_a(x) = a·x is injective; right cancellation means that g_a(x) = x·a is injective. On finite sets, injective maps are automatically surjective, so each f_a and g_a becomes surjective as well. Surjectivity supplies exactly the needed x and y for every choice of a and b: for any b, there exists x with a·x = b and exists y with y·a = b. That matches the lemma’s condition, forcing S to be a group.

So the practical message is crisp: in finite semigroups, “cancellation on both sides” isn’t merely a property of groups—it is a characterization of them.