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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
Revision as of 17:45, 19 April 2012
, 17:45, 19 April 2012→6.6. Basic Notions of Group Theory: markup
A binary operation or LOC <math>*\!</math> on <math>X\!</math> is '''commutative''' if and only if <math>x*y = y*x\!</math> for every <math>x, y \in X.\!</math>
A binary operation or LOC <math>*\!</math> on <math>X\!</math> is '''commutative''' if and only if <math>x*y = y*x\!</math> for every <math>x, y \in X.\!</math>
A '''semigroup''' consists of a nonempty set with an associative LOC on it. On formal occasions, a semigroup is introduced by means a formula like <math>\underline{X} = (X, *),\!</math> read to say that <math>\underline{X}\!</math> is the ordered pair <math>(X, *).\!</math> This form specifies <math>X\!</math> as the nonempty set and <math>*\!</math> as the associative LOC. By way of recalling the extra structure, this specification underscores the name of the set <math>X\!</math> to form the name of the semigroup <math>\underline{X}.\!</math> In contexts where there is only one semigroup being discussed, or where the additional structure is otherwise understood, it is common practice to call the semigroup by the name of the underlying set. In contexts where more than one semigroup is formed on the same set, one can use notations like <math>\underline{X}_i = (X, *_i)\!</math> to distinguish them.
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A "unit element" in a semigroup X = <X, *> is an element e in X such that x*e = x = e*x for all x C X. In other words, a unit element is a two sided identity element. If a semigroup X has a unit element, then it is unique, since if e' is also a unit element, then e' = e'*e = e.
A "unit element" in a semigroup X = <X, *> is an element e in X such that x*e = x = e*x for all x C X. In other words, a unit element is a two sided identity element. If a semigroup X has a unit element, then it is unique, since if e' is also a unit element, then e' = e'*e = e.