Talk:Reflexive Closure of Relation Compatible with Operation is Compatible

A preordering as defined on this site is already reflexive. It is defined (you will see it if you click on the link for Definition:Preordering) as being a relation which is reflexive and transitive.

If you are using a definition of preorder which does not match this, then (a) you need to define exactly what it is, and (b) you need to give it another name, because what you're talking about is not it. --prime mover (talk) 15:32, 6 January 2013 (UTC)


 * Sorry, I was for some reason convinced it said transitive and antisymmetric (although surprised by that). That's a rather different matter and I will rename pronto. --Dfeuer (talk) 16:00, 6 January 2013 (UTC)


 * You will find there is already a series of pages which tread the same ground that you are working on. In particular, the relation between $\prec$ and $\preceq$ is established rigorously and there should be absolutely no need for the construct $\preceq = \prec \cup \Delta_S$ which makes my brain hurt.


 * If you use the term "strict ordering" to refer to an antireflexive transitive antisymmetric relation (which is I believe what you are requiring $\prec$ to be, you will save yourself (and the reader) a lot of unnecessary work. --prime mover (talk) 16:07, 6 January 2013 (UTC)


 * Lord_Farin invoked your name earlier when I was looking for info on relations compatible with operations and you didn't speak up, so I really hope that wasn't all duplicate work. Where might I find the info on the relationship between $\prec$ and $\preceq$? --Dfeuer (talk) 16:17, 6 January 2013 (UTC)


 * I didn't speak up because I didn't notice the posting. Besides, as it was discussing what I believed to be preorderings not strict orderings, I didn't recognise the work as being stuff that I had already addressed.


 * Try looing in order theory. Ordering is Strict Ordering Union Diagonal Relation is a place to start. Then that will give you an idea of what terminology to use. --prime mover (talk) 16:22, 6 January 2013 (UTC)


 * I don't care about that. That's just the tail end of the work I've been doing. I'm much more concerned about all the rest of the proofs I created in Category:Compatible Relations. --Dfeuer (talk) 16:26, 6 January 2013 (UTC)


 * I'll leave you to explore then. I have other stuff I'm doing at the moment. --prime mover (talk) 16:50, 6 January 2013 (UTC)

Maybe it's a good idea to introduce the "reflexive closure" (possibly the "antireflexive interior", but I just made that up) of a relation, and refer to that in place of the strings of symbols cropping up everywhere now. --Lord_Farin (talk) 22:16, 6 January 2013 (UTC)


 * Definition:Reflexive Closure and Definition:Reflexive Reduction are what are being talked about here.


 * I completely agree with L_F here: $\preceq = \prec \cup \Delta_S$ is unremittingly ugly. Better (if not using reflexive closure) to actually write something like: "Let $\preceq$ be defined as $\prec \cup \Delta_S$" - anything less is egregiously appalling. --prime mover (talk) 22:25, 6 January 2013 (UTC)


 * Thanks for locating those. Incidentally, it seems they are up for a rewrite to current schemes. --Lord_Farin (talk) 22:27, 6 January 2013 (UTC)


 * I greatly prefer "Let $\preceq$ be the reflexive closure of $\prec$" over your alternative; so much more elegant. --Lord_Farin (talk) 22:31, 6 January 2013 (UTC)


 * I agree, but I was offering an alternative in case it is decided by the page author that the language of Definition:Reflexive Closure and Definition:Reflexive Reduction are inappropriate on this page for any reason. --prime mover (talk) 22:53, 6 January 2013 (UTC)

Also, please note that I didn't craft the Order Theory section and its relatives so that I have no sight on what is resp. isn't up already. I'm afraid you're on your own here. I'll try to trace your steps, though. --Lord_Farin (talk) 22:17, 6 January 2013 (UTC)

I'll be glad to use reflexive closure and reflexive reduction. I'd love to have a name for a transitive, antisymmetric relation. Is subordering available? --Dfeuer (talk) 23:37, 6 January 2013 (UTC)


 * There's no name for it because there's no need for it. Orderings and strict orderings are useful in themselves, but this neither-one-thing-not-another object, while easy to create, formulate and understand, doesn't have a great deal of direct use. If you need to discuss a transitive antisymmetric relation, afraid you'll have to call it that. I don't think it's helpful to coin a term here unless you can show it's a useful concept in its own right by, e.g. publishing papers pointing out important results based on it. --prime mover (talk) 06:20, 7 January 2013 (UTC)