Magmas and Semigroups

A prealgebraic structure starts with a set and an operation on that set.

Magma

A magma is a set S together with a binary operation * : S x S -> S. The only requirement is closure: combining two elements of S stays in S.

Example: The integers with subtraction form a magma, because a - b is an integer for any integers a, b.

Semigroup

A semigroup is a magma whose operation is associative. Associativity means (a * b) * c = a * (b * c) for all a, b, c in S.

Example: The natural numbers with addition form a semigroup.

Why this matters

Many later structures require associativity before they introduce identities, inverses, or distributive laws. Semigroups are the first stable step in that direction.