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Semigroup–Quasigroup Structure
{{< euler-diagram >}}
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 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 quasigroup is a set with a binary operation such that for all , the equations
each have unique solutions .
A quasigroup with an identity element is called a loop.
A loop is a quasigroup with an identity element. That is, a set with a binary operation such that:
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:
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.