16 Dec 2012

winterkoninkje: shadowcrane (clean) (Default)

Friday I turned in the last of my papers, and last night I turned in the last of my grades. Which means I'm free for a whole two weeks or so. Which is good because for the last couple weeks I've been wanting desperately to blog as a means of escaping all those due dates. So now I get to flush some of the backlog from this semester. In particular, there's a cluster of posts I've been meaning to put up for a while, a sort of Intro to Maths for mathematicians, as it were. To save y'all's friendspages I'll spread these posts out over the next few days.

columbicubiculomania — The compulsion to stick things into pigeonholes. (Jim Matisoff 1990)

Ever since stumbling upon that fabulous word I've wanted to spread its popularity. As a geek with certain obsessive–compulsive tendencies, I'm a bit prone to columbicubiculomania; or rather, as the theoretician that I am, I'm prone to the dual of columbicubiculomania. I don't care so much about the pigeons, they can take care of themselves. But I do have a certain compulsion to design and arrange pigeonholes such that whenever someone feels the columbicubiculocompulsion, they'll have the right places to stuff all their pigeons into. Part of this is also tied up in me trying to figure out (or rather, to convey) where exactly I situate myself in the vast sea of competing academic fields. As the perennial outsider, I'm more interested (it seems) in seeing how everything is connected than are those stuck on the inside.

And so, over the past few years I've been accumulating the jargon from different subfields of mathematics and organizing them into a cohesive whole. In particular, for the next few posts, I'm concerned with the terminology for algebraic objects. Too often I read papers with practitioners of one algebraic field (e.g., group theory, ring theory,...) getting unwittingly caught up in another field and subsequently using such outrageous terms as "semigroup with identity" or "monoid without identity". Because of this ridiculousness, a lot of mathematicians and computer scientists end up not realizing when the thing they're studying has already been studied in depth under another name. So let's see all those names and how they connect. Let's produce a cartography of mathematical objects.

Perhaps the easiest place to start is one of the latter maps I produced. Binary relations are ubiquitous in all areas of mathematics, logic, and computer science. Typically we don't care about all binary relations, we only care about relations with particular properties. Of course, the interaction between properties is nontrivial since properties A and B together can entail that property C must hold. Thus, a map of binary relations is helpful for keeping track of it all. This map requires relatively little explanation, which is why I started here.

All the common properties of interest are defined at the top, and color coded for use in the map. And, constructivist that I am, I've been sure to distinguish between strong and weak versions of the properties (which collapse in the classical setting). The arrowheads in the map are for keeping track of when we're talking about P, anti-P, or co-P (for some particular property P). And the big black dot is the starting point of the map (i.e., a binary relation with no special properties). The dashed lines indicate when some property will follow for free. So, for example, there's no name for a transitive irreflexive relation since trans+irrefl entails antisymmetry and trans+irrefl+antisym is a strict partial order. And the black lines are for when the named object requires some peculiar property that doesn't show up all over the place. These conventions for dashed and black lines are common to all my maps.

Once you're familiar with my conventions, the only thing really left unstated on this map is that the apartness relation used in the definition of a strongly connected binary relation is actually a tight apartness. The constructive significance of apartness relations vs equality relations is something I'll have to talk about another time.

Towards the top of the map we start veering away from binary relations per se and start heading into order theory. Because the complexities of order theory are a bit different from the complexities of binary relations, I've chosen to break them up into different maps. As yet, I haven't actually made the map for order theory, but I'll get around to it eventually. Lattice theory and domain theory also gets tied up in all this (in the order theory particularly). I've started work on a lattice theory map, but haven't finished it just yet.

This map is released under Creative Commons Attribution-ShareAlike 3.0. Any questions? Anything I missed? Anything that needs further explanation?

June 2017

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