Tour:Some variations of group

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WHAT YOU NEED TO DO: Read, and understand, all the definitions presented below. These involve variations on the notion 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.