Apparently I messed up my link to the map of ring theory last time. It's fixed now, but the fact that noone commented on it means noone's reading this blog anymore (or those who do don't care). Ah well. Regardless, it's time to finish up this little series for now.
Anyone who's hung around me long enough knows that I'm a great fan of semirings. Partly this is because of their ubiquity in natural language processing, but mostly it's because they're the intersection of two of the most powerful and ubiquitous abstractions in mathematics. Semirings simultaneously generalize rings (by not necessarily having negation) and bounded lattices (by not necessarily being commutative nor having both operators distribute). Therefore, they generalize both arithmetic and ordering, two of the core components of mathematics. Also the fact that the natural numbers do not form a ring is a strong case for taking semirings seriously. I don't have a map this time, instead I have an extensive list of common examples. The examples fall into about half a dozen common patterns.
- Natural number-like semirings
- Integer-like rings
- Tropical and Viterbi semirings
- Bounded distributive lattices
- Boolean rings
- Regular languages
- Square matrices over a semiring
I'm sure you're already familiar with the natural numbers, integers, etc., so I won't spend much time talking about them. The one thing worth mentioning here, though, is what the heck that last column is for. It's for *-semirings (i.e., closed semirings), not to be confused with *-(semi)rings (i.e., (semi)rings with involution). Jacques Carette introduced me and Russell O'Connor to them, and Russell has written a bit about them. Since he does a good job of explaining them, I won't talk about them just now. But, because they're a bit obscure, I made sure to indicate what the asteration operation is when it exists. The one especially notable point here is that the Alexandroff compactification of the reals is no longer a ring! (Another reason to take semirings seriously.) We give up the ring-ness of the reals to obtain closed-ness and compact-ness.
The tropical and Viterbi semirings are a sort of cross between arithmetic and ordering, and you've probably never heard of them before. They are, however, rather common in computer science. They tend to arise whenever we use dynamic programming in order to find the "best" of some set of things: e.g., the most likely tag sequence, the shortest path, the longest path, etc. This is what part of speech tagging, matrix chain multiplication, and Dijkstra's algorithm are all about. And recognizing that these are semirings is what leads to generalizations like forward–backward and inside–outside, to say nothing of the Gauss–Jordan–Floyd–Warshall–McNaughton–Y
The last major class I'll talk about are the bounded distributive lattices. The reason to discuss these is just to point out that this includes all Boolean algebras (hence classical logic and subset inclusion hierarchies), all Heyting algebras (hence intuitionistic logic), and all bounded totally ordered sets.
This chart is released under Creative Commons Attribution-ShareAlike 3.0. Any questions? Anything I missed? Anything that needs further explanation?