I am currently writing a paper where I need to define an antimatroid, which is a combinatorial structure that can be axiomatized in several equivalent ways.
In the axiom system I am using, an antimatroid consists of a finite set and a collection of subsets of , obeying several conditions. One of these conditions is:
For any with and in , we can find a chain , such that all the lie in and is a singleton.
Of course, I could equivalently say
For any with and and in , we can find in such that .
For any with and in , we can find in such that and is a singleton.
Same as , with the roles of and switched.
My preference is for . What is yours? More generally, should definitions include the fewest conditions possible, so that the statement “ is a boojum” is easy to check, or the most, so that it is easy to apply?