# Talk:List of order structures in mathematics

WikiProject Mathematics (Rated List-class, Mid-priority)
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Field:  Foundations, logic, and set theory

## do not commute

This article is one of the stranger maths articles here: "do not commute" under what operation? Perhaps I'm failing to understand. — Preceding unsigned comment added by 217.158.210.179 (talk) 21:10, 29 March 2003‎ (UTC)
You're absolutely right. The person who wrote this is writing nonsense; he seems unaware of the difference between a set and an ordered tuple. Michael Hardy 21:34 Mar 29, 2003 (UTC)

Some context made clear what the original write meant by "does not communte" or "does not permute", but, as a mathematician who is good at understanding others' mathematical writing even when their terminology differs from what I am familiar with, I nonetheless found those terms confusing at first, and I have to think others would have as well. The article is now much better than it was; other articles that already existed address the topics. Michael Hardy 21:48 Mar 29, 2003 (UTC)

How about if we simply redirect to partial order and be done with it? AxelBoldt 00:43 Mar 30, 2003 (UTC)

How about a little bit of layman's terms here; there has to be a simple way of explaining this. I am really struggling with these highly rigorous definitions... Steve

## Different definition

I thought that an ordered set meant something where the order of the elements mattered (like a sequence, i.e. putting the elements in a different order results in a different ordered set), but where duplicates are not allowed (like a set). --208.80.119.67 (talk) 21:27, 9 June 2011 (UTC)

## Structure of the list

There was some discussion of this list on Wikipedia_talk:WikiProject_Mathematics. Three comments:

1) Should this list have some kind of structure? For example, many of the items on this list (semiorders, interval orders) are probably best understood as special kinds of posets, while others (well-orderings) probably aren't. Should the list be structured to reflect this?

2) Regarding types of posets: it seems to me that e.g. semiorder belongs on this list and Eulerian poset may not; is there a precise way to state this distinction? Or do we think adjective poset belongs on this list for every value of adjective that gives a blue link?

3) One thing this list doesn't currently address is more concrete orderings: lexicographic order, monomial order, etc. Do they belong here? If not, where do they belong? Other possibilities (and other pages that this list should avoid just being a duplicate of) are List of order theory topics and Glossary of order theory. --JBL (talk) 19:09, 11 March 2013 (UTC)

As I remarked below, I wanted to list different axiomatizations/concepts of the intuitive idea of an "order" (total, partial, weak, etc) and not concrete examples of orders of these types (which could go in a different list). One sort-of-soapboxy purpose is as a counter to the sort of thinking you display: it is not true that posets encapsulate all of the kinds of orderings one would want to think about. Well-orderings are perfectly well understood as a kind of total order with additional structure, just as a semiorder can be understood as a partial order with additional structure. But cyclic orders, for instance, aren't even binary relations; nevertheless they are a type of order. Preorders are strict generalizations of posets. Weak orders can be axiomatized as posets with a transitive incomparability relation, but they don't have to be; they can equally well be axiomatized as preorders in which the equivalence classes are totally ordered, or as total orders on the sets in a set partition. And although I haven't included them in the list, I think of antimatroids as an order-theoretic generalization of partial orders: a partial order specifies a collection of total orders (its linear extensions), and an antimatroid also specifies a collection of total orders (its basic words); the collections that can be represented by antimatroids are a strict superset of the collections that can be represented by finite partial orders. —David Eppstein (talk) 20:26, 11 March 2013 (UTC)
Re "The sort of thinking [I] display", I'm not sure you read my comment quite right. As I said, some types of orders (e.g., well-orders) are probably not best thought of as partial orders (even thought they are). On the other hand, I don't know that there's an independent study of semiorders (e.g.) that studies properties that aren't also interesting for all posets. So a natural structure to this list might put semiorders under posets (but probably shouldn't put well-orders under posets). I was asking whether the list should be structured in some way that reflects this, and if not then how. --JBL (talk) 21:43, 11 March 2013 (UTC)
It's a reasonable question, but when we look at it in detail it's a bit problematic. E.g. are weak orders a type of partial order (strict weak orders) or a type of preorder (total preorders)? That depends not on the conceptual idea of what a weak order represents, but rather on the detailed choice of axiomatization. Are semiorders a generalization of weak orders that should be listed as a subtopic of them, or are weak orders a specialization of semiorders that should be listed as a subtopic of them? Something like map of lattices that shows the inclusion relations between these different kinds of orders might make more sense, or at least maybe we could sort it into a linear extension of the inclusion partial order. —David Eppstein (talk) 22:16, 11 March 2013 (UTC)

## Title of the list

“List of types of ordered set” is a gibberish. I would prefer “List of order structures” – is looks more scientific. Incnis Mrsi (talk) 19:56, 11 March 2013 (UTC)

There was a discussion at WT:WPM about a list of individual ordered sets; I wanted to distinguish the subject from that. Your proposed title doesn't do that. And I have no idea what you mean when you say "is a gibberish" and "looks more scientific". Is "types" not the right word, and if so what is (kinds? classes?). —David Eppstein (talk) 20:20, 11 March 2013 (UTC)
I would reluctantly accept “List of order(ed) types of a set”, but not the present form. Since we agreed that an “ordered set” has not a universal meaning, the only way to understand a “type of an ordered set” becomes the “type of a set, which is ordered”. I do not understand what “type of a set” supposed to mean, in general. Which type, or types, have sets of integers and real numbers, for example? What is the type, or types, of the empty set? Note that English is not my native language, though. Incnis Mrsi (talk) 20:38, 11 March 2013 (UTC)
But they are not types of sets. They are types of mathematical object of the form (set, ordering information). I would like to call such an object an "ordered set". What would you like to call it? —David Eppstein (talk) 20:41, 11 March 2013 (UTC)
"List of types of ordered sets" would be better grammatically. ("List of types of car" is broken in the same way.) --JBL (talk) 21:39, 11 March 2013 (UTC)