Changes

One says that <math>\underline{X}\!</math> is '''written additively''' to mean that a plus sign <math>(+)\!</math> is used instead of a star for the LOC.  In this case, the notation <math>x + y\!</math> indicates a value in <math>X\!</math> called the '''sum''' of <math>x\!</math> and <math>y.\!</math>  This involves the further conventions that the unit element is written as a zero, <math>0,\!</math> and may be called the '''zero element''', while the inverse of an element <math>x\!</math> is written as <math>-x,\!</math> and may be called the '''negative''' of <math>x.\!</math>  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.

One says that <math>\underline{X}\!</math> is '''written additively''' to mean that a plus sign <math>(+)\!</math> is used instead of a star for the LOC.  In this case, the notation <math>x + y\!</math> indicates a value in <math>X\!</math> called the '''sum''' of <math>x\!</math> and <math>y.\!</math>  This involves the further conventions that the unit element is written as a zero, <math>0,\!</math> and may be called the '''zero element''', while the inverse of an element <math>x\!</math> is written as <math>-x,\!</math> and may be called the '''negative''' of <math>x.\!</math>  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.

: 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 written additively, 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> 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 monoid written additively, 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> 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>

: In a group written additively, 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.

The '''induced action''' of a function <math>f : X\to Y\!</math> on the cartesian power <math>X^n\!</math> is the function <math>f' : X^n \to Y^n\!</math> defined by:

The '''induced action''' of a function <math>f : X\to Y\!</math> on the cartesian power <math>X^n\!</math> is the function <math>f' : X^n \to Y^n\!</math> defined by:

: <math>f'(x_1, \ldots, x_n) ~=~ (f(x_1), \ldots, f(x_n)).\!</math>

: <math>f'(x_1, \ldots, x_n) ~=~ (fx_1, \ldots, fx_n).\!</math>

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Usually, <math>f'\!</math> is regarded as the ''natural'', ''obvious'', ''tacit'', or ''trivial'' extension that <math>f : X \to Y\!</math> possesses in the space of functions <math>X^n \to Y^n,\!</math> and is thus allowed to go by the same name as <math>f.\!</math> This convention, assumed by default, is expressed by the formula:

Usually, f' is regarded as the "obvious", "trivial", or "tacit" extension that f : X >Y possesses in the space of functions Xn >Yn, and is thus allowed to go by the same name.  This convention, assumed by default, is expressed by the formula:

f(<x1, ..., xn>) = <f(x1), ..., f(xn)>.

: <math>f(x_1, \ldots, x_n) ~=~ (fx_1, \ldots, fx_n).\!</math>

A "relation homomorphism" from an n place relation P c Xn to an n place relation Q c Yn is a mapping between the underlying sets, h : X >Y, whose induced action h : Xn >Yn preserves the indicated relations, taking every element of P to an element of Q.  In other words:

A "relation homomorphism" from an n place relation P c Xn to an n place relation Q c Yn is a mapping between the underlying sets, h : X >Y, whose induced action h : Xn >Yn preserves the indicated relations, taking every element of P to an element of Q.  In other words: