Changes

: The <math>n^\text{th}\!</math> '''multiple''' of an element <math>x\!</math> in a semigroup <math>\underline{X} = (X, +, 0),\!</math> for integer <math>n > 0,\!</math> is notated as <math>nx\!</math> and defined as follows.  Proceeding recursively, for <math>n = 1,\!</math> let <math>1x = x,\!</math> and for <math>n > 1,\!</math> let <math>nx = (n-1)x + x.\!</math>

: The <math>n^\text{th}\!</math> '''multiple''' of an element <math>x\!</math> in a semigroup <math>\underline{X} = (X, +, 0),\!</math> for integer <math>n > 0,\!</math> is notated as <math>nx\!</math> and defined as follows.  Proceeding recursively, for <math>n = 1,\!</math> let <math>1x = x,\!</math> and for <math>n > 1,\!</math> let <math>nx = (n-1)x + x.\!</math>

: The "nth multiple" of x in a monoid X = <X, +, 0>, for integer n > 0, is defined the same way for n > 0, letting 0x = 0 when n = 0.

: <math>n^\text{th}\!</math> '''multiple''' of <math>x\!</math> in a monoid <math>\underline{X} = (X, +, 0),\!</math> for integer <math>n \ge 0,\!</math> is defined the same way for <math>n > 0,\!</math> letting <math>0x = 0\!</math> when <math>n = 0.\!</math>

: The "nth multiple" of x in a group X = <X, +, 0>, for any integer n, is defined the same way for n > 0, letting nx = ( n)( x) for n < 0.

: <math>n^\text{th}\!</math> '''multiple''' of <math>x\!</math> in a group <math>\underline{X} = (X, +, 0),\!</math> for any integer <math>n,\!</math> is defined the same way for <math>n \ge 0,\!</math> letting <math>nx = (-n)(-x)\!</math> for <math>n < 0.\!</math>

: A group X = <X, +, 0> is "cyclic" if and only if there is an element g C X such that every x C X can be written as x = ng for some n C Z.  In this case, an element such as g is called a "generator" of the group.

: A group <math>\underline{X} = (X, +, 0)\!</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 = ng\!</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.

====Version 2====

====Version 2====

: In a semigroup <math>\underline{X} = (X, +, 0),\!</math> the <math>n^\text{th}\!</math> '''multiple''' of an element <math>x\!</math> is notated as <math>nx\!</math> and defined for every positive integer <math>n\!</math> in the following manner.  Proceeding recursively, let <math>1x = x\!</math> and let <math>nx = (n-1)x + x\!</math> for all <math>n > 1.\!</math>

: In a semigroup <math>\underline{X} = (X, +, 0),\!</math> the <math>n^\text{th}\!</math> '''multiple''' of an element <math>x\!</math> is notated as <math>nx\!</math> and defined for every positive integer <math>n\!</math> in the following manner.  Proceeding recursively, let <math>1x = x\!</math> and let <math>nx = (n-1)x + x\!</math> for all <math>n > 1.\!</math>

: In a monoid <math>\underline{X} = (X, +, 0),\!</math> <math>nx\!</math> is defined for every non-negative integer <math>n\!</math> by letting <math>0x = 0\!</math> and proceeding the same way for <math>n > 0.\!</math>

: In a monoid <math>\underline{X} = (X, +, 0),\!</math> the multiple <math>nx\!</math> is defined for every non-negative integer <math>n\!</math> by letting <math>0x = 0\!</math> and proceeding the same way for <math>n > 0.\!</math>

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

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

: A group <math>\underline{X} = (X, +, 0),\!</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 = ng\!</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} = (X, +, 0)\!</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 = ng\!</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.