Changes

Let's see how this remark applies to the order-preserving property of the "number of" mapping <math>v : S \to \mathbb{R}.</math>  For any pair of absolute terms <math>x\!</math> and <math>y\!</math> in the syntactic domain <math>S,\!</math> we have the following implications, where <math>^{\backprime\backprime}-\!\!\!<\!^{\prime\prime}</math> denotes the logical subsumption relation on terms and <math>^{\backprime\backprime}\!\!\le\!^{\prime\prime}</math> denotes the ''less than or equal to'' relation on the real number domain <math>\mathbb{R}.</math>

Let's see how this remark applies to the order-preserving property of the "number of" mapping <math>v : S \to \mathbb{R}.</math>  For any pair of absolute terms <math>x\!</math> and <math>y\!</math> in the syntactic domain <math>S,\!</math> we have the following implications, where <math>^{\backprime\backprime}-\!\!\!<\!^{\prime\prime}</math> denotes the logical subsumption relation on terms and <math>^{\backprime\backprime}\!\!\le\!^{\prime\prime}</math> denotes the ''less than or equal to'' relation on the real number domain <math>\mathbb{R}.</math>

: ''x'' –< ''y'' &rArr; ''vx'' =< ''vy''

x ~-\!\!\!< y & \Rightarrow & vx \le vy

\end{array}</math>

Equivalently:

Equivalently:

: ''x'' –< ''y &rArr; [''x''] =< [''y'']

x ~-\!\!\!< y & \Rightarrow & [x] \le [y]

It is easy to see that nowhere near all of the distinctions that make up the structure of the ordering on the left hand side will be preserved as one passes to the right hand side of these implication statements, but that is not required in order to call the map ''v'' "order-preserving", or what is also known as an "order morphism".

Nowhere near the number of logical distinctions that exist on the left hand side of the implication arrow can be preserved as one passes to the linear ordering of real numbers on the right hand side of the implication arrow, but that is not required in order to call the map <math>v : S \to \mathbb{R}</math> ''order-preserving'', or what is known as an ''order morphism''.

===Commentary Note 11.19===

===Commentary Note 11.19===