Changes

It is customary to use a number of abbreviations and conventions in discussing semigroups, monoids, and groups.  A system <math>\underline{X} = (X, *)\!</math> is given the adjective ''commutative'' if and only if <math>*\!</math> is commutative.  Commutative groups, however, are traditionally called ''abelian groups''.  By way of making comparisons with familiar systems and operations, the following usages are also common.

It is customary to use a number of abbreviations and conventions in discussing semigroups, monoids, and groups.  A system <math>\underline{X} = (X, *)\!</math> is given the adjective ''commutative'' if and only if <math>*\!</math> is commutative.  Commutative groups, however, are traditionally called ''abelian groups''.  By way of making comparisons with familiar systems and operations, the following usages 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.

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.

The '''<math>n^\text{th}\!</math> power''' of <math>x\!</math> in a semigroup <math>\underline{X} = (X, \cdot),\!</math> for a positive integer <math>n,\!</math> is notated as <math>x^n\!</math> and defined as follows.  Proceeding recursively, for <math>n = 1,\!</math> let <math>x^1 = x,\!</math> and for <math>n > 1,\!</math> let <math>x^n = x^{n-1} \cdot x.\!</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 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>

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: 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>

The "nth power" of x in a monoid X = <X, ., 1>, for natural number n, is defined the same way for n > 0, letting x0 = 1 when n = 0.

The "nth power" of x in a group X = <X, ., 1>, for arbitrary integer n, is defined the same way for n > 0, letting xn = (x 1)( n) for n < 0.

: 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>

A group X = <X, ., 1> is "cyclic" if and only if there is an element g C X such that every x C X can be written as x = gn 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}\!</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.

2. When one says that X is "written additively" it means that a plus sign "+" is used instead of a star "*" for the LOC.  In this case, the notation "x + y" indicates a value in X called the "sum" of x and y.  This involves the further conventions that the unit element is written as a zero "0", and may be called the "zero element", while the inverse of an element x is written as " x", and may be called the "negative of x".  Usually, but not always, this manner of presentation is reserved for commutative systems and abelian groups.  In the additive idiom, the following definitions of "multiples", "cyclic groups", and "generators" are also common.

When one says that X is "written additively" it means that a plus sign "+" is used instead of a star "*" for the LOC.  In this case, the notation "x + y" indicates a value in X called the "sum" of x and y.  This involves the further conventions that the unit element is written as a zero "0", and may be called the "zero element", while the inverse of an element x is written as " x", and may be called the "negative of x".  Usually, but not always, this manner of presentation is reserved for commutative systems and abelian groups.  In the additive idiom, the following definitions of "multiples", "cyclic groups", and "generators" are also common.

The "nth multiple" of x in a semigroup X = <X, +>, for integer n > 0, is notated as "nx" and defined as follows.  Proceeding recursively, for n = 1, let 1x = x, and for n > 1, let nx = (n 1)x + x.

The "nth multiple" of x in a semigroup X = <X, +>, for integer n > 0, is notated as "nx" and defined as follows.  Proceeding recursively, for n = 1, let 1x = x, and for n > 1, let nx = (n 1)x + x.