necessary or sufficient properties for 'greater than' ( >)

I wonder which properties are necessary or sufficient to define
correctly -i.e. in a way which makes sense mathematically- 'greater
than' for a given set. I have been searching a bit already on the
Internet and thus far I came to the following conclusion (but I'm
rather unsure about it):

1. not (a<a)
2. if a<b then not(b<a)
3. if a<b and b<c then a<c
4. if a<b and c>0 then ac<bc

can anyone on this list comment whether these four conditions are
sufficient and/or necessary?

thanks

nukey
.

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