Magmas and Semigroups

Wed, 31 Dec 2025 00:00:00 +0000

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.

Wed, 31 Dec 2025 00:00:00 +0000

Monoids add an identity element, and homomorphisms describe structure-preserving maps between monoids.

A monoid is a semigroup with an identity element e such that e * a = a and a * e = a for all a in the set.

Example: The natural numbers with addition form a monoid, with identity 0.

Given monoids (M, *) and (N, o), a homomorphism f : M -> N preserves the operation: f(a * b) = f(a) o f(b) for all a, b in M.