Definition talk:Relation Compatible with Operation

Added initial sketch of proof limited to preorders.

I haven't looked at the reference but if there is a proof of equivalence without that precondition I would like to see it.

Concept seems to inherently require reflexivity and I don't know how to prove without transitivity.

I have only seen the concept of a relation being compatible with an operation in the context of congruences (equivalences) and orders. If that is the case, restriction to preorders is not problematic.

So I am guessing the original claim without the limitation to preorders was an error.


 * Probably. My bad. --prime mover 14:10, 20 June 2011 (CDT)


 * This section has now been tidied up, corrected and rationalised. --prime mover 03:43, 23 March 2012 (EDT)


 * Unless I just haven't explored this section enough I may have to introduce another distinction here. As concatenation is "right-compatible" with the lexicographical ordering of words. But not "left-compatible". Best to do so after the actual proof though. --Jshflynn 16:08, 15 August 2012 (UTC)

I disagree on that last one. Generally, unless I feel the stuff is too intuitive to be bothered (like some pointwise stuff in the Measure Theory section), the definition should come before the use. I may not be consistent in applying this but mostly that'll be due to awkward ordering in books.

It be explicated however that I *do* think it is a good idea to separate both the cases (although I don't see why lex.order wouldn't be left-compatible...) --Lord_Farin 16:57, 15 August 2012 (UTC)


 * Darn it. I wish I would take a moments extra thought before posting stuff like this on here. You are correct L_F (and quite clearly so). Now I'm scratching my head trying to think of any possible use of the distinction. --Jshflynn 17:50, 15 August 2012 (UTC)


 * The prefix relation seems to work. That is, we say $w \preceq w'$ if $w'$ starts with $w$. It is left-compatible (obvious), but not right-compatible (for adding stuff at the end generally messes up prefix-ness). --Lord_Farin 17:55, 15 August 2012 (UTC)