Changes

One says that <math>\underline{X}\!</math> is ''written multiplicatively'' to mean that a raised dot <math>(\cdot)\!</math> or concatenation is used instead of a star for the LOC.  In this case, the unit element is commonly written as an ordinary algebraic one, <math>1,\!</math> while the inverse of an element <math>x\!</math> is written as <math>x^{-1}.\!</math>  The multiplicative manner of presentation is the one that is usually taken by default in the most general types of situations.  In the multiplicative idiom, the following definitions of ''powers'', ''cyclic groups'', and ''generators'' are also common.

One says that <math>\underline{X}\!</math> is ''written multiplicatively'' to mean that a raised dot <math>(\cdot)\!</math> or concatenation is used instead of a star for the LOC.  In this case, the unit element is commonly written as an ordinary algebraic one, <math>1,\!</math> while the inverse of an element <math>x\!</math> is written as <math>x^{-1}.\!</math>  The multiplicative manner of presentation is the one that is usually taken by default in the most general types of situations.  In the multiplicative idiom, the following definitions of ''powers'', ''cyclic groups'', and ''generators'' are also common.

: In a semigroup, the <math>n^\text{th}\!</math> power of an element <math>x\!</math> is notated as <math>x^n\!</math> and defined for a positive integer <math>n\!</math> in the following manner.  Proceeding recursively, let <math>x^1 = x\!</math> and let <math>x^n = x^{n-1} \cdot x\!</math> for all <math>n > 1.\!</math>

: In a semigroup, the <math>n^\text{th}\!</math> power of an element <math>x\!</math> is notated as <math>x^n\!</math> and defined for every positive integer <math>n\!</math> in the following manner.  Proceeding recursively, let <math>x^1 = x\!</math> and let <math>x^n = x^{n-1} \cdot x\!</math> for all <math>n > 1.\!</math>

: In a monoid, <math>x^n\!</math> is defined for every non-negative integer <math>n\!</math> by letting <math>x^0 = 1\!</math> and proceeding the same way for <math>n > 0.\!</math>

: In a monoid, <math>x^n\!</math> is defined for every non-negative integer <math>n\!</math> by letting <math>x^0 = 1\!</math> and proceeding the same way for <math>n > 0.\!</math>

: In a group, <math>x^n\!</math> is defined for every integer <math>n\!</math> by letting letting <math>x^n = (x^{-1})^{-n}\!</math> for <math>n < 0\!</math> and proceeding the same way for <math>n > 0.\!</math>

: In a group, <math>x^n\!</math> is defined for every integer <math>n\!</math> by letting <math>x^n = (x^{-1})^{-n}\!</math> for <math>n < 0\!</math> and proceeding the same way for <math>n > 0.\!</math>

: A group <math>\underline{X}\!</math> is '''cyclic''' if and only if there is an element <math>g \in X\!</math> such that every <math>x \in X\!</math> can be written as <math>x = g^n\!</math> for some <math>n \in \mathbb{Z}.\!</math>  In this case, an element such as <math>g\!</math> is called a '''generator''' of the group.

: A group <math>\underline{X}\!</math> is '''cyclic''' if and only if there is an element <math>g \in X\!</math> such that every <math>x \in X\!</math> can be written as <math>x = g^n\!</math> for some <math>n \in \mathbb{Z}.\!</math>  In this case, an element such as <math>g\!</math> is called a '''generator''' of the group.