17 Dec 2012

winterkoninkje: shadowcrane (clean) (Default)

Continuing the thread from last time, let's move on from relations and consider a map of individual binary operations. In a lot of ways this is even simpler than the binary relations from last time, though the map requires a bit more explanation. This time, rather than having definitions at the top, they're given as labels on the arcs. Arcs in the same color denote the same property, dashed lines represent things you get for free, and black lines are for the odd things; all just like last time.

Most of the study of individual binary operations falls under group theory, which forms the core of this map. The one interesting thing here is that if you have at least monoid structure (i.e., have an identity element) then the uniqueness of inverses follows from having the presence of inverses. However, for semigroups which are not monoids, these two properties differ. This'll come up again next time when we start talking about rings and fields.

Off to the left we veer into lattices. And to the right we get the crazy stuff that comes from non-associative algebra. Quasigroups and loops are somewhat similar to groups in that they have an invertible structure, but unfortunately they don't have associativity. It turns out, there's a whole hierarchy of almost-but-not-quite-associative properties, which is shown on the second page. The strongest property you can get without being fully associative is Moufang, which can be phrased in four different ways. Below this we have left- and right-Bol (if you have both the Bols then you have Moufang). Below that we have alternativity where you choose two of three: left-alternativity, right-alternativity, and flexibility. Below that, of course, you can have just one of those properties. And finally, at the bottom, power associativity means that powers associate (and so "powers" is a well-formed notion) but that's it.

As I said, there's not a whole lot here, but I needed to bring this one up before getting into ring-like structures. This map is released under Creative Commons Attribution-ShareAlike 3.0. Any questions? Anything I missed? Anything that needs further explanation?

April 2017

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