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A '''monoid''' is a semigroup with a unit element. Formally, a monoid <math>\underline{X}\!</math> is an ordered triple <math>(X, *, e),\!</math> where <math>X\!</math> is a set, <math>*\!</math> is an associative LOC on the set <math>X,\!</math> and <math>e\!</math> is the unit element in the semigroup <math>(X, *).\!</math>
A '''monoid''' is a semigroup with a unit element. Formally, a monoid <math>\underline{X}\!</math> is an ordered triple <math>(X, *, e),\!</math> where <math>X\!</math> is a set, <math>*\!</math> is an associative LOC on the set <math>X,\!</math> and <math>e\!</math> is the unit element in the semigroup <math>(X, *).\!</math>
An '''inverse''' of an element <math>x\!</math> in a monoid <math>\underline{X} = (X, *, e)\!</math> is an element <math>y \in X\!</math> such that <math>x*y = e = y*x.\!</math> An element that has an inverse in <math>\underline{X}\!</math> is said to be '''invertible''' (relative to <math>*\!</math> and <math>e\!</math>). If <math>x\!</math> has an inverse in <math>\underline{X},\!</math> then it is unique to <math>x.\!</math> To see this, suppose that <math>y'\!</math> is also an inverse of <math>x.\!</math> Then it follows that:
An "inverse" of an element x in a monoid X = <X, *, e> is an element y C X such that x*y = e = y*x. An element that has an inverse in X is said to be "invertible" (relative to * and e). If x C X has an inverse, then it is unique to x. To see this, suppose that y' is also an inverse of x. Then it follows that:
y' = y'*e = y'*(x*y) = (y'*x)*y = e*y = y.
: <math>y' ~=~ y'*e ~=~ y'*(x*y) ~=~ (y'*x)*y ~=~ e*y ~=~ y.\!</math>
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A "group" is a monoid all of whose elements are invertible. That is, a group is a semigroup with a unit element in which every element has an inverse. Putting all the pieces together, then, a group X = <X, *, e> is a set X with a binary operation * : XxX >X and a designated element e that is subject to the following three axioms:
A "group" is a monoid all of whose elements are invertible. That is, a group is a semigroup with a unit element in which every element has an inverse. Putting all the pieces together, then, a group X = <X, *, e> is a set X with a binary operation * : XxX >X and a designated element e that is subject to the following three axioms:
