Unital Magma
A magma with an identity element
A unital magma is a magma equipped with an identity element satisfying
for all .
Examples
- Any monoid, loop, or group
- A set with a non-associative operation that still has an identity element
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 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 loop is a quasigroup with an identity element. That is, a set with a binary operation such that:
- Identity: There exists with for all
- Divisibility: For all , the equations and have unique solutions
A loop that is also associative is a group. The smallest non-associative loop has 5 elements.
A group is a set together with a binary operation such that:
- (Associativity) For all , .
- (Identity) There exists an element such that for all , and .
- (Inverses) For every there exists an element such that and .
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.