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.

<pre>

: 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>

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 monoid X = <X, +, 0>, for integer n > 0, is defined the same way for n > 0, letting 0x = 0 when n = 0.

: 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>

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.

: 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>

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.

Mathematical systems, like the Rs and Xs encountered above, are seldom comprehended in perfect isolation, but need to be viewed in relation to each other, as belonging to families of comparable systems.  Systems are compared by finding or making correspondences between them, and this can be formalized as a task of setting up and probing various types of mappings between the sundry appearances of their objective structures.  This requires techniques for exploring the spaces of mappings that exist between families of systems, for inquiring into and demonstrating the existence of specified types of functions between them, plus technical concepts for classifying and comparing their diverse representations.  Therefore, in order to compare the structures of different objective systems and to recognize the same objective structure when it appears in different phenomenal or syntactic disguises, it helps to develop general forms of comparison that can organize the welter of possible associations between systems and single out those that represent a preservation of the designated forms.

Mathematical systems, like the Rs and Xs encountered above, are seldom comprehended in perfect isolation, but need to be viewed in relation to each other, as belonging to families of comparable systems.  Systems are compared by finding or making correspondences between them, and this can be formalized as a task of setting up and probing various types of mappings between the sundry appearances of their objective structures.  This requires techniques for exploring the spaces of mappings that exist between families of systems, for inquiring into and demonstrating the existence of specified types of functions between them, plus technical concepts for classifying and comparing their diverse representations.  Therefore, in order to compare the structures of different objective systems and to recognize the same objective structure when it appears in different phenomenal or syntactic disguises, it helps to develop general forms of comparison that can organize the welter of possible associations between systems and single out those that represent a preservation of the designated forms.