Changes

An order relation is typically defined by a set of axioms that determines its properties.  Since we have frequent occasion to view the same set in the light of several different order relations, we often resort to explicit specifications like <math>(X, <_1),\!</math> <math>(X, <_2),\!</math> and so on, to indicate a set with a given ordering.

An order relation is typically defined by a set of axioms that determines its properties.  Since we have frequent occasion to view the same set in the light of several different order relations, we often resort to explicit specifications like <math>(X, <_1),\!</math> <math>(X, <_2),\!</math> and so on, to indicate a set with a given ordering.

A map ''F'' : (''X''₁,&nbsp;&lt;₁) &rarr; (''X''₂,&nbsp;&lt;₂) is ''order-preserving'' if and only if a statement of a particular form holds for all ''x'' and ''y'' in (''X''₁,&nbsp;&lt;₁), specifically, this:

A map <math>F : (X_1, <_1) \to (X_2, <_2)</math> is ''order-preserving'' if and only if a statement of a particular form holds for all <math>x\!</math> and <math>y\!</math> in <math>(X_1, <_1),\!</math> specifically, this:

: ''x'' &lt;₁ ''y'' &rArr; ''Fx'' &lt;₂ ''Fy''

| <math>x <_1 y ~\Rightarrow F(x) <_2 F(y).</math>

The action of the "number of" map ''v'' : (''S'', &lt;₁) &rarr; ('''R''', &lt;₂) has just this character, as exemplified by its application to the case where ''x'' = ''f'' = "frenchman" and ''y'' = ''m'' = "man", like so:

The action of the "number of" map ''v'' : (''S'', &lt;₁) &rarr; ('''R''', &lt;₂) has just this character, as exemplified by its application to the case where ''x'' = ''f'' = "frenchman" and ''y'' = ''m'' = "man", like so: