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: 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 $a*b=a*c⟹b=c$. Similarly $a\in S$ is termed right cancellative if $b*a=c*a⟹b=c$. 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 right invertible.
• Group: A group is a monoid where every element is invertible.