Originally Posted by

**ymar** I'm not sure that this is the right place to post my question, but other places don't seem right either.

While dealing with a certain highly irregular algebraic structure with two operations, say $\displaystyle \circ$ and $\displaystyle \square,$ and a partial order, say $\displaystyle \leq,$ I was trying to find any relation between the two operations. An obvious guess was the distributive property but it's not there. I thought I would try the following property

$\displaystyle a\circ(b\square c)\leq (a\circ b)\square (a\circ c).$

It turns out it doesn't hold for all $\displaystyle a,b,c$ either. However, I wrote a program to see how many of all possible triples $\displaystyle (a,b,c)$ satisfy the above formula. It seems, experimentally, that most of them do. The structure is unambiguously given by its size $\displaystyle n$. It seems to me that for $\displaystyle n$ approaching infinity the probability of the formula being satisfied for random $\displaystyle a,b,c$ diverges to 1, quite quickly at that. I haven't proven it and I don't know if it isn't well beyond my skills, but this is not why I'm writing this.

I found the idea of a structure "almost having" a property (in this probabilistic sense) intriguing. I would be interested to know if you have encountered any research in this direction. Do people do this? Is it useful?