What do we mean by permissible transformations in types of attributes-:nominal,ordinal,interval,ratio?

I am studying data mining and I stumbled upon types of attributes.

They are

1. Nominal

2. Ordinal

3. Interval

4. Ratio

Data mining book by Tan,Steinbech,Kumar says Permissible transformations for-:

1. nominal-: any one to one mapping, eg a permutation of values.

2. Ordinal-: new_value=f(old_value). An order preserving change of values.

3. Interval-: new_value=a*old_value+b

4. Ratio-: new_value=a*old_value

I tried making sense of this, but could not really make sense what is this trying to say.

What I know?

1. Nominal attributes-:

It provides enough information to distinguish one object from another.

eg-: gender, zipcodes, employee id numbers, jersey number of players.

Here we can't average the jersey number of player values and find something meaningful.

The numbers even they are integers, no mathematical operations can be performed except =,≠.

2. Ordinal-:

Ordidnal values provide enough information to order the objects.

eg-: grades, {good,better,best}

Operation ,

If ram's percentage=90%, mohan's percentage=45%, we can't say Ram is 2 times good as Mohan.

3. Interval-:

Here the difference between values are meaninful. There is no absolute 0 so we can't take the ration of 2 measurements. eg-: temperature in celsius, fahrenheit, time of day etc.

We can't say 10 AM is twice as long as 5 AM. But we can say

0-10 AM interval=10 hrs

0-5 AM interval=5 hrs

We can say interval between 0-10 AM is twice as long as interval between 0-5 AM. This is because 0 AM doesn't mean absence of any time.

When we say 0 F we don't mean zero heat.

Also, 100 degree F isn't twice as 50 degree F.

4. Ratio-:

For ratio variables, both differences and ratio are meaninful. Eg-: temperature in kelvin,mass, monetary quantities etc. It has meaningful zero point. If someone's income is 0, then income is 0 unlike 0 AM which means 12 midnight.