The question http://mathoverflow.net/questions/48117/multi-inequality was closed.

As Ryan Budney pointed out, multiple inequalities occur in the Simplex algorithm and in describing polyhedra. Thanks Ryan for pointing that out to me.

To clarify, I was asking about single expressions like 2+3i(> <)1+4i in which each side is related by a multi-inequality symbol. I have edited the question to make this clearer.

I thought it would be interesting to consider taking an inequality expression and replacing the inequality symbol with a multi-inequality symbol, and let the variables be members of non-real sets.

Perhaps those closing knew what I meant and closed anyway, in which case I'll have to accept that. Could this question please be considered for reopening ?

The question has the form "Has anyone seen this mathematical concept before?" But I don't see a new mathematical concept: the concept of finite systems of inequalities has been around for centuries. I just see a proposed new notation for the concept. It seems reasonable to close questions of the form "Has this notation been used before?" because 1) they are quite localized in nature, perhaps not of interest outside of the user of the notation, 2) in most cases the answer is probably going to be "no", but how do you conclusively establish that a certain notation has never before been used?, and 3) such questions could be endlessly generated.

It sounds like you wish to ask about something beyond notation, but honestly whatever else you may have in mind is far from clear. That is why you were asked to edit the question before asking it to be reopened. As far as I can see you have barely edited the question at all. Remember that the burden is always on the questioner to make himself/herself clear. Please put a little more effort into this.

I think the answer to your question is that multi-inequality notation brings no added value. Multi-inequalities are just a bunch of inequalities with "and" statements stuck between them. There's no compelling reason to have a general-purpose notation based on multi-inequalities as they're so closely related to things we already use.

There is a distinction here between a partially ordered set and a multi-ordered set. As far as I know a tree can have a partial order, and so can the complex numbers, but the complex numbers being component-based with each of the components having a total order means that the complex numbers have a natural multi-order i.e. a collection of relations that cover the whole set, so that any two complex numbers are related by one of the relations (<<), (<>), (<=),( ><),(>>),(>=),(=<),(=>),(==). The real inequality a<b is transformed into -a>-b. The question is if you have a multi-order then what transform do you need to turn say w(<<)z into Tw(>>)Tz, or Sw(<>)Sz, Rw(><)Rz etc. There probably aren't any such transforms for the complex numbers, but maybe there are in other situations, so the question becomes are there any component-based objects where the inequalities between the components change in a predictable way when the object as a whole is transformed.

My motivation for this is the same as for my earlier questions about cyclic order relations http://mathoverflow.net/questions/25505/cyclic-order-relation-in-zn and about set inclusions at http://www.google.com/buzz/114134834346472219368/FAYeqjLsKYK/Much-of-hard-analysis-relies-extensively-on: namely to look for ungrazed pastures - there is a huge literature on inequalities, so if the concept of inequality is widened all the questions about inequalities can be asked again and perhaps give rise to non-real constants http://mathoverflow.net/questions/24740/non-real-constants

@Roy Maclean: In fact I just looked up multi-order on mathscinet. I got some hits, most of which refer to the order of a differential equation and one in mathematical logic. Sieving these out, I found the same paper you reference above. In fact this is the unique hit for "multi-ordered set" on mathscinet.

The fact that you're interested in this particular kind of structure does clarify your question a bit.

But I still don't understand the part about the complex numbers. The multi-order you define is not compatible with the product of complex numbers in any reasonable way. So what do you mean by a "transform"? One way to transform (<,<) into (>,>) is simply to send (x,y) to (-x,-y). But this seems trivial.

If you can clarify this point, you might have a "real question" and I would consider voting to reopen.

An example would be the product algebra R x R x ... R with coordinatewise multiplication. Then as I said above, you can track the change on multi-orders just by looking at the signs on the components.

Again, this is not a very interesting example. I'm trying to show you that your question is under-determined. It's some kind of question, but not the sort of precise, specific question that is appropriate on MO. "Is there an algebra over R with a multi-order [you will have to precisely define this] satisfying the following properties...?" would be roughly the right format for a MO question. To me, it seems that you're throwing out rather vague ideas and hoping that someone else can make them precise. That's what friends are for, not MO.