"Grandmother"
Monday, 4. August 2008, 00:44:24
Since I mentioned it in the forums ...
My graduate work in Mathematics was in a rather specialized discipline, called "ordered permutation groups" (though my disssertation was actually on a different topic). You have a set - typically the real numbers but any totally ordered set will do - and then you have this collection of functions acting on your set elements. There are three operations defined on these functions: supremum (at each point it uses the larger of the two functions' values), infimum (the smaller) and basic composition of functions.
That's the most general example, what are called "lattice-ordered groups" or "l-groups". My advisor had proved that any l-group could be considered as an ordered permutation group, and of course wrote the textbook we used ...
Anyway, there were 4 basic examples of ordered permutation groups. One was just all strictly increasing functions on the real numbers, another was basic addition (which was a totally-ordered group), the third was the set of all periodic functions (meaning there is a fixed number n for which f(x+n) = f(x)+n for all x), and the fourth example ...
The fourth example had a similar definition to the third example, except in reverse. Rather than saying there was one specific number n for which f(x+n) = f(x)+n, the definition was that each function f had such a number n (non-zero) but that the number depended on the function f.
Since the definitions were similar, the last two examples were given similar nicknames - the third example was "grandfather" and the fourth example was "grandmother. But "grandmother" had another name as well ...
What's so special about all this? Well, a lot of the theorems in lattice-ordered groups had one exception. Yeah, that's right, we had hundreds of theorems which applied to all l-groups except "grandmother", therefore "grandmother" was also referred to as "the pathological 2-transitive l-group".
You've all heard the expression "There's an exception to every rule"? Well, not only is there an exception, in the case of lattice-ordered groups we even knew exactly what that exception was. Makes things easy in that you know where to look for the exception, but is also disconcerting in that this one specific example is always the exception (hence the use of the term "pathological").
So ... what's your "grandmother" like?
My graduate work in Mathematics was in a rather specialized discipline, called "ordered permutation groups" (though my disssertation was actually on a different topic). You have a set - typically the real numbers but any totally ordered set will do - and then you have this collection of functions acting on your set elements. There are three operations defined on these functions: supremum (at each point it uses the larger of the two functions' values), infimum (the smaller) and basic composition of functions.
That's the most general example, what are called "lattice-ordered groups" or "l-groups". My advisor had proved that any l-group could be considered as an ordered permutation group, and of course wrote the textbook we used ...
Anyway, there were 4 basic examples of ordered permutation groups. One was just all strictly increasing functions on the real numbers, another was basic addition (which was a totally-ordered group), the third was the set of all periodic functions (meaning there is a fixed number n for which f(x+n) = f(x)+n for all x), and the fourth example ...
The fourth example had a similar definition to the third example, except in reverse. Rather than saying there was one specific number n for which f(x+n) = f(x)+n, the definition was that each function f had such a number n (non-zero) but that the number depended on the function f.
Since the definitions were similar, the last two examples were given similar nicknames - the third example was "grandfather" and the fourth example was "grandmother. But "grandmother" had another name as well ...
What's so special about all this? Well, a lot of the theorems in lattice-ordered groups had one exception. Yeah, that's right, we had hundreds of theorems which applied to all l-groups except "grandmother", therefore "grandmother" was also referred to as "the pathological 2-transitive l-group".
You've all heard the expression "There's an exception to every rule"? Well, not only is there an exception, in the case of lattice-ordered groups we even knew exactly what that exception was. Makes things easy in that you know where to look for the exception, but is also disconcerting in that this one specific example is always the exception (hence the use of the term "pathological").
So ... what's your "grandmother" like?
WIDGETIZE
Felix Pleşoianu # 4. August 2008, 09:27
JD # 9. August 2008, 01:37