Effect monoids are mathematical structures (like e.g. groups, rings, vector spaces, etc.) that axiomatize the unit interval, [0,1], or, in other words, probabilities, in much the same way that groups axiomatize symmetries. An effect monoid consists of a multiplication, and a partial addition (50% + 25% makes sense, but a probability of 50% + 55% does not) that need to obey certain axioms. Not only the unit interval, [0,1], is an effect monoid, but also, for example, any Boolean algebra, such as {0,1}.

When dealing with such an abtract stucture, one naturally wonders if all instances can be classified somehow. A few years back, we worked out that that any effect monoid that is ‘directed complete’ (=suprema of directed sets exist) must be of the form B ⨁ [0,1]_C(X), where B is a complete Boolean algebra, and [0,1]_C(X) is the set of all continuous functions X →[0,1], where X is some (extremally disconnected) compact Hausdorff space. From this one could argue that [0,1] and {0,1} are really the only possible probabilities, at least, when directed completeness is assumed.

For the weaker condition of ω-completeness (suprema of countable directed sets exist) something similar can be shown. Given the existing representation theorems for e.g. commutative C*-algebras one would expect that the condition can be weakened further to something like ‘metric completeness’. Such a result we did not manage to obtain back then. We did not even find a workable definition of metric completeness for effect monoids — but perhaps you might!

Please read this paper if you think this might be might an interesting challenge.

In PubHubs hubs, users are identified by a (shortened) hash, like a3g-a48, to, in principle, prevent impersonation. “In principle”, because it’s unlikely that you’d remember another user’s hash, let alone notice when it’s changed. That’s why we’re considering using an ‘identicon’ like the one used by GitHub (and others), but we see two problems:

1. How to prevent users from being assigned an identicon they consider offensive? (For example, one having the colors of an opposing soccer team.)
2. Is it feasible for an attacker obtain an identicon that’s close enough to another person’s identicon, by registering a whole bunch of user accounts.