User talk:Lord Farin/Sandbox/Definition:Lattice

I think this is a good start, but I have two thoughts:
 * 1. Since the name "lattice" comes from its shape as an ordered structure, and the ordered structure is extremely easy to describe, I think that definition should come first.
 * Disagree - putting the constructivist approach first seems far more sensible, as it indicates how to build one of these things. Def 2 posits the existence of such a structure which already has those properties. --prime mover (talk) 16:31, 23 December 2012 (UTC)
 * Disagree with disagreement: commutative, associative, idempotent operations satisfying absorption laws aren't any more obviously constructable. --Dfeuer (talk) 16:53, 23 December 2012 (UTC)
 * Disagree with dis etc.: commutativity, associativity, idempotent operations satisfying absorption laws aren't even mentioned (except in the first definition they are explicitly shown). --prime mover (talk) 17:15, 23 December 2012 (UTC)
 * Aside from the absorption laws, which are stated but not named, the axioms are brought in through the definition of semilattice, and, through that, the definition of semigroup. --Dfeuer (talk) 18:05, 23 December 2012 (UTC)
 * Whatever. You're the boss. --prime mover (talk) 18:25, 23 December 2012 (UTC)

I'd say I'm the boss as I took the initiative :). I tend to agree with Dfeuer on the grounds that Def 2 is shorter and indeed gives the more important "order-theoretic" approach (though really both defs use an ordering). 't Is however precisely this interplay between order theory and algebra that makes this area so interesting and surprisingly rich, and ultimately this all is merely a matter of personal preference (or democracy if you wish).
 * 2. I think it would be good to include (in addition) expanded definitions that don't rely on semilattice, join semilattice, and meet semilattice. --Dfeuer (talk) 16:21, 23 December 2012 (UTC)
 * Excellent idea. We probably want to put a page together called "Lattice Axioms" like we do with various other abstract algebraic concepts - one in progress now, as I type ... --prime mover (talk) 16:31, 23 December 2012 (UTC)

I have that residing in the pipeline but haven't gotten round to it yet (nasty loose course work ends took most of my "free" time this weekend). Hopefully I'll find time tomorrow. --Lord_Farin (talk) 22:11, 23 December 2012 (UTC)


 * Both points accounted for, I hope. The fully equational characterization I got from my BSc. thesis. --Lord_Farin (talk) 17:59, 24 December 2012 (UTC)
 * I hope you appreciate the way I styled the axioms (in this way, the duality principle is immediately apparent). --Lord_Farin (talk) 18:01, 24 December 2012 (UTC)

Incompatibility
Currently, Defs 1 and 4 include that a lattice is bounded (has greatest and smallest elts) while 2 and 3 don't. Cursory research reveals that these concepts are referred to as a bounded lattice and lattice, resp., though many authors impose boundedness and still call the result a lattice. Seems there is more work to do before this can move to main. --Lord_Farin (talk) 20:04, 24 December 2012 (UTC)


 * Definition 4 just needs a separate page, for bounded lattice, and we need "see also" links from "lattice" to "bounded lattice", "complete lattice", and anything else important.. --Dfeuer (talk) 21:13, 24 December 2012 (UTC)