Changes

===Commentary Note 11.18===

===Commentary Note 11.18===

There is a comment that I ought to make on the concept of a ''structure preserving map'', including as a special case the idea of an ''order-preserving map''.  It seems to be a peculiarity of mathematical usage in general — at least, I don't think it's just me — that "preserving structure" always means "preserving ''some'', not of necessity ''all'' of the structure in question".  People sometimes express this by speaking of ''structure preservation in measure'', the implication being that any property that is amenable to being qualified in manner is potentially amenable to being quantified in degree, perhaps in such a way as to answer questions like "How structure-preserving is it?".

There is a comment that I ought to make on the concept of

a "structure preserving map", including as a special case

the idea of an "order-preserving map".  It seems to be a

peculiarity of mathematical usage in general -- at least,

I don't think it's just me -- that "preserving structure"

always means "preserving 'some', not of necessity 'all',

of the structure in question".  People sometimes express

this by speaking of "structure preservation in measure",

the implication being that any property that is amenable

to being qualified in manner is potentially amenable to

being quantified in degree, perhaps in such a way as to

answer questions like "How structure-preserving is it?".

Let's see how this remark applies to the order-preserving property of

Let's see how this remark applies to the order-preserving property of the "number of" mapping ''v''&nbsp;:&nbsp;''S''&nbsp;&rarr;&nbsp;'''R'''.  For any pair of absolute terms ''x'' and ''y'' in the syntactic domain ''S'', we have the following implications, where "–<" denotes the logical subsumption relation on terms and "=<" is the "less than or equal to" relation on the real number domain R.

the "number of" mapping 'v' : S -> R.  For any pair of absolute terms

x and y in the syntactic domain S, we have the following implications,

where "-<" denotes the logical subsumption relation on terms and "=<"

is the "less than or equal to" relation on the real number domain R.

x -< y =>  'v'x =< 'v'y

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

Equivalently:

Equivalently:

x -< y =>  [x] =< [y]

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

It is easy to see that nowhere near all of the distinctions that make up

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".

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".

===Commentary Note 11.19===

===Commentary Note 11.19===