# Definition talk:Well-Ordering

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Conflict:

On this page in the definition section on line 3: "has a **minimal element** under $\preceq$"

On the well-founded page in the definition section on line 2: "has a **smallest element**."

-Jshflynn 29/6/12

- Good call. Replaced "Minimal" with "Smallest". In a woset, of course, the two are the same, as a woset is a toset and Minimal Element in Toset is Unique and Smallest. But it's good to be consistent.

- BTW here's a technique you might want to use: when writing a post in a talk page, press the button above the edit pane which has a squiggle in it. It will add two dashes followed by four tildes. The latter will be replaced by the mediawiki s/w with your username (known as your "sig"). Like what follows here for me: --prime mover 20:56, 29 June 2012 (UTC)

## recent change

Why the recent change? IMO it's useful to link to well-founded.

Is the current definition of well-founded inapplicable in this context or something? --prime mover (talk) 08:01, 5 April 2013 (UTC)

- That should be fixed by importing the other definition from Definition:Well-Ordered Set, and then gutting that page. Two options for defining are:

- Total ordering which is a well-founded ordering (which should
*not*just be called "well-founded", because that mostly means something else, as far as I can tell) - Ordering such that every nonempty subset has a smallest element.

- Total ordering which is a well-founded ordering (which should

- Note that the ordering $\{ (x,x), (y,y) \}$ on $\{ x, y \}$ is well-founded ($x$ and $y$ are both minimal), but not a well-ordering (no smallest element). --Dfeuer (talk) 13:35, 5 April 2013 (UTC)

- I'm going to have to revert all the changes you are doing today because they are in danger of changing the validity of the exising definitions. As usual there are no sources quoted. --prime mover (talk) 18:13, 5 April 2013 (UTC)

- Okay I think I can see where it all went wrong now ... part of the original confusion between minimal and smallest elements. Okay, I'm with you now.

- Still think it's worth separating out "well-founded set" and "well-founded relation" perhaps transcluding them both in a parent page "well-founded".

- Okay, sorry for interrupting - but can you put "SourceReview" templates on all pages where you change stuff around so I have a flag to go and check and make sure it's all pointing in the right direction? --prime mover (talk) 18:55, 5 April 2013 (UTC)

- Well-founded set isn't even on the table yet, as far as I know, but it probably will be (as a special case of well-founded relation). Well-founded ordering/ordered set and well-founded relation/relational structure (currently and I believe legitimately also called foundational) are on the table, and neither is a special case of the other. I will gladly put in some templates. --Dfeuer (talk) 19:00, 5 April 2013 (UTC)

- "Well-founded set" as in "well-founded relational structure" then, whatever - in the same way you say "well-ordered set" to mean a relational structure where the relation is a well-ordering, innit? --prime mover (talk) 19:05, 5 April 2013 (UTC)