Talk:Total order

message: I think I'm sure I've seen one, but can't think of an example just right now. 67.198.37.16 (talk) 18:11, 16 January 2019 (UTC)
replies: text: A total order is reflexive (), while a strict total is not (x \not < x). So a total order is never a strict total order. The article explains in detail the relationship between the two concepts, namely that there are equivalent (a total order defines a strict total order, a strict total order defines a total order, and these two operations are inverses one to other). D.Lazard (talk) 23:24, 16 January 2019 (UTC)
thread_title:  Example of total order that is not a strict total order? 

message: From the examples section.
: The unique order on the empty set, ∅, is a total order.
: ...
: The natural numbers comprise the smallest totally ordered set with no upper bound.
: The integers comprise the smallest totally ordered set with neither an upper nor a lower bound.
The linked article on Upper and lower bounds only talks about upper and lower bounds of subsets of an ordered set.  If we interpret the above remarks about the natural numbers and integers as statements about these sets as subsets of themselves, then the remarks are not correct.  ∅ (viewed as a subset of itself) is the smallest unbounded totally ordered set, there being no element of ∅ capable of serving as a bound.  On the other hand, if we view these sets as proper subsets of some other totally-ordered superset, then ∅ is not an unbounded set, being bounded by every element of its superset.  But in this case whether the natural numbers or integers are bounded depends upon the nature of the total order of their superset.  For example if the natural numbers are viewed as an initial segment of some larger ordinal, then they are bounded above.
Either way, the statements as currently written are not universally true.
Minimally the the word "non-empty" needs to be inserted, which I will do shortly.  Beyond that, either the statements in this article should to be revised to make it clear that the sets are being viewed as subsets of themselves, or if it is commonplace in current mathematical discourse to refer to bounds of ordered sets without reference to subsets, then the article on Upper and lower bounds should be edited to explain this usage.  — Preceding unsigned comment added by 93.97.176.246 (talk) 05:56, 11 March 2020 (UTC)
thread_title:  Smallest bounded sets 

message: I think it is possible to add a remark that the approach 
"We can work the other way and start by choosing < as a transitive trichotomous binary relation;" 
is equivalent in the classical logic, but is stronger in the intuitionistic logic. (So one who uses it is advised to choose the "Strict Total Order" in one's definitions.)  — Preceding unsigned comment added by Georgydunaev (talk • contribs) 08:52, 6 September 2020 (UTC)
thread_title:  Intuitionistic comment 

message: When viewing the article on mobile the binary relations table shows first, fully expanded. I’m not sure how best to fix this (and I am on mobile) but it seems an unhelpful way to begin new readers. — HTGS (talk) 08:22, 15 June 2021 (UTC)
replies: text: Being not about the subject of this article, this table does not belong to the lead. I have moved it to the section "Related structures, where it is more appropriate. D.Lazard (talk) 12:08, 15 June 2021 (UTC)
thread_title:  Binary relations table 

message: A total order is currently being defined as follows: "That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a (reflexive).
# If a \leq b and b \leq c then a \leq c (transitive).
# If a \leq b and b \leq a then a = b (antisymmetric).
# a \leq b or b \leq a (strongly connected, formerly called total)."
This is redundant – 1. follows from 4. if you substitute a for b (since it doesn't say "for all distinct a, b").
Joriki (talk) 19:26, 16 February 2023 (UTC)
replies: text: . I added a remark in the lead about that. - Jochen Burghardt (talk) 08:58, 17 February 2023 (UTC)
thread_title:  The current definition is redundant 

message: I expected the section "Strict and non-strict total orders" to contain a definition of a non-strict total order. It currently only defines only strict total orders and then discusses relationship between strict and non-strict total orders. Including a definition would help readers' understandings of core concepts in this article. Bradknox (talk) 21:18, 11 May 2023 (UTC)
replies: text: An occurence of "strict" was omitted. Otherwise, the definition of a (non strict) total order is given in the preceding section D.Lazard (talk) 21:45, 11 May 2023 (UTC)
replies: text: I added an appropriate sentence to the article. - Jochen Burghardt (talk) 12:42, 13 May 2023 (UTC)
thread_title:  Non-strict total order should be defined