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A set with a binary operation (no other axioms)
A magma is a set together with a binary operation . No additional axioms are required—the operation need not be associative, commutative, or have an identity.
This is the most general algebraic structure with a single binary operation. All semigroups, monoids, and groups are magmas.
A semigroup is a set together with an associative binary operation , meaning that for all ,
Semigroups are the weakest common algebraic setting in which repeated products are unambiguous (associativity lets you omit parentheses). A monoid is a semigroup with an identity element, and a group is a monoid in which every element has an inverse.
A monoid is a semigroup together with an element (called an identity element) such that for every ,
Monoids generalize groups by dropping the requirement that elements have inverses. Many monoids arise from composition of endomorphisms (self-maps).
A group is a set together with a binary operation such that:
Equivalently, a group is a monoid in which every element is invertible. Much of group theory studies subgroups and structure-preserving maps called group homomorphisms.