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>

: <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 <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>

: <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>

: The <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>

====Version 2====

====Version 2====