Polarities (Part 4)

In Part 2, I explained some stuff you can do with graphs whose edges are labeled by elements of a rig. Remember, a rig is like a ring, but it might not have negatives. A great example is the boolean rig, whose elements are truth values:

The addition in this rig is ‘or’ and the multiplication is ‘and’.

So what’s the point? If we have a graph with edges labeled by booleans, we can use them to describe whether one vertex affects another. For example:

Here directly affects because the first edge is labeled but does not directly affect because the second edge is labeled To figure out whether affects , we multiply the booleans and using ‘and’. We get so does not affect

But what about this example?

Now has the ability to affect along two edges. To figure out the total effect we should add the booleans and using ‘or’. We get so does affect

I should formalize what I’m doing here a bit more carefully, so it applies to arbitrary finite graphs with edges labeled by booleans. But right now I want to talk about using other rigs to label edges. We can use any rig, which we call the rig of polarities. Then a graph with edges labeled by elements of this rig is called a causal loop diagram.

We can get rigs from hyperrings, and I want to explain how. In Part 3 I explained ‘hyperrings’, which are like rings, but where addition can be multivalued. A really good one is the hyperring of signs:

If we use these as polarities, this means that directly affects in a positive way:

In other words, increasing the quantity tends to increase Similarly, an edge labeled with means one quantity directly affects another in a negative way, while an edge labeled with means no direct effect.

In the hyperring multiplication is defined in the obvious way, and addition is also obvious except when we add and What’s the sum of a positive number and a negative number? It could be positive, negative or zero! So we make addition multivalued, write it as and say

All this is nice, which suggests we should study hyperrings of polarities. But then I noticed that we can get a rig from this hyperring… and this rig may do all the jobs we need done.

If we take all the elements of the hyperring and start adding, subtracting and multiplying to our heart’s content, what do we get? We get a bunch of subsets of . The elements we start with are

but in this game we identify these with the corresponding singleton sets

What other subsets of do we get by adding, subtracting and multiplying these singletons? We get other singletons except in one case. Namely, the sum of and equals this:

These four subsets are all that we get:

Now here’s the cool part. In a hyperring we can extend addition, multiplication, and subtraction to operations on subsets. I explained how this works for addition last time, but it works the same way for the other operations. And in our example here, the collection of the above 4 subsets forms a rig under multiplication and addition! The other subsets just never come up.

So, we get a little rig with 4 elements I’ll call

Here or indeterminate is short for the set In this language we say the sum of and is This nicely captures the idea that when you add up two quantities, and all you know is that one is positive and the other is negative, the sign of the result is indeterminate.

Here’s the addition table for this rig

+0–i+++ii0+0–i–i––iiiiii

Here is the multiplication table:

+0–i++0–i00000––0+iii0ii

It’s possible this rig is a perfectly good substitute for our original hyperring when we’re using its elements to label edges in a causal loop diagram. Only time will tell: we need to develop the theory of causal loop diagrams while keeping this question in mind.

But there are also a math question here. How generally can we get a rig like this from a hyperring? Let me try to tackle that.

Given any hyperring we can extend its addition and multiplication to the collection of nonempty subsets of as follows. To multiply two nonempty subsets we define

To add them, we define

Remember, the sum of two elements of a hyperring is already a subset of that hyperring.

Now, you might hope these operations make into a rig. But that’s not always true! The reason—or at least one reason—is that not every hyperring obeys the double distributive law

whereas this law holds if is a rig.

I don’t know any other problems, so I’ll make this guess. It should be easy to prove if true, but I’ll call it a ‘conjecture’ to motivate people—including myself—to prove or disprove it.

Conjecture. Suppose is a doubly distributive hyperring. Then becomes a rig if we define addition and multiplication as above, with as its additive unit and as its multiplicative unit.

If this is true, we can then define a subrig of the way I did in the example where is the hyperring of signs. Namely, let be the smallest subrig of containing all the singletons where This subrig may be ‘good enough for causal loop diagrams’.

This would be a systematic way to get the rig

from the hyperring of signs

That would be great! But even if the conjecture needs to be tweaked, at least I know that there exists this rig in this special case. It’s already used for polarities in causal loop diagrams. I’m just trying to understand it better—for example, I’ve never heard anyone say it’s a rig, or take advantage of the fact that it’s a rig. We need some mathematicians to get into this game, make the math nice and shiny, and make it more powerful!