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A magma with an identity element
A unital magma is a magma equipped with an identity element satisfying
Unlike a monoid, a unital magma need not have an associative operation.
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.
A monoid is a semigroup with an identity element , meaning that for every ,
Monoids generalize groups by dropping the requirement that elements have inverses. Many monoids arise from composition of endomorphisms (self-maps).