A magma, or groupoid, is a set with a single binary operation.
These can be defined as an ordered pair \((s,\odot )\) where \(s\) is the set, and \(\odot\) is the binary operation.
If \(a\) and \(b\) are in \(s\), then \(a\odot b\) is also in \(s\).
The following are magmas:
Natural numbers and addition
\(n\times n\) matrices with determinants other than \(0\)
Natural numbers above \(0\) and addition
Integers and addition
Rational numbers and division
\(\{-1, 1\}\) and multiplication
The following are not magmas:
Natural numbers up to \(10\) and addition
A semigroup is a magma whose binary operation is associative.
The following are semigroups:
Natural numbers and addition
\(n\times n\) matrices with determinants other than \(0\)
Natural numbers above \(0\) and addition
Integers and addition
\(\{-1, 1\}\) and multiplication
The following are not semigroups:
Rational numbers and division
Natural numbers up to \(10\) and addition
A monoid is a semigroup with an identity element.
The following are monoids:
Natural numbers and addition
\(n\times n\) matrices with determinants other than \(0\)
Integers and addition
\(\{-1, 1\}\) and multiplication
The following are not monoids:
Natural numbers above \(0\) and addition
Rational numbers and division
Natural numbers up to \(10\) and addition
A group is a monoid where there is an inverse operation for the binary operation.
The following are groups:
Integers and addition
\(n\times n\) matrices with determinants other than \(0\)
\(\{-1, 1\}\) and multiplication
The following are not groups:
Natural numbers above \(0\) and addition
Rational numbers and division
Natural numbers and addition
Natural numbers up to \(10\) and addition