Tour:Some variations of group


 * Magma: A magma is a set $$S$$ with a binary operation $$*: S \times S \to S$$. There is no condition of associativity, there is no requirement that an identity element exist, and there is no condition for inverses of any kind to exist.
 * Semigroup: This is a magma where the associativity condition is satisfied. For any $$a,b,c \in S$$, we have $$a * (b * c) = (a * b) * c$$
 * Neutral element (also termed identity element): An element $$e \in S$$ is termed left neutral if $$e * a = a$$ for all $$a$$, right neutral if $$a * e = a$$ for all $$a$$. $$e$$ is termed neutral if it is both left and right neutral. A neutral element is also termed an identity element.
 * Monoid: A monoid is a semigroup with a neutral element.
 * Cancellative element: An element $$a \in S$$ is termed left cancellative if $$a * b = a * c \implies b = c$$. Similarly $$a \in S$$ is termed right cancellative if $$b * a = c * a \implies b = c$$. An element is termed cancellative if it is both left and right cancellative.
 * Invertible element: In a magma with neutral element $$e$$, an element $$a$$ is said to be left invertible if there exists $$b$$ such that $$b * a = e$$, and right invertible if there exists $$c$$ such that $$a * c = e$$. If there exists a $$b$$ such that $$a * b = b * a = e$$, the element is termed invertible.
 * Group: A group is a monoid where every element is invertible.