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However, not for every strict weak order there is a corresponding real function. For example, there is no such function for the lexicographic order on Rn.
But is seems easy to derive that there is such a function for Qn × R. We can add that if there is a source.-- Patrick 11:31, 21 May 2007 (UTC)
…and weak order gets redirected here? How is this different from making strict partial order a main article that contains a definition of partial order as a variant of the primary topic? I propose turning things around (for weakness, not for partiality).— PaulTanenbaum ( talk) 03:21, 18 March 2008 (UTC)
Currently there's a pretty picture on the page (like a snowflake or poinsettia) that does not illustrate what a strict weak order is. (It illustrates something else, which is not entirely irrelevant; I'm not saying the picture is misleading, just that it's not the one most important picture we need to have on this page.)
It would be awesome if we could just show the graph of a strict weak order on some smallish set. Strict weak orders are really very easy to understand; I think this is a topic where a good picture would be worth all the rest of the words on the page. — Jorend ( talk) 04:31, 13 December 2009 (UTC)
I've made two illustrations, that are inspired by drawings in this PDF (pages 23 and 26):
The assignment of the permutations to the vertices is not the one from the permutohedron ( compare), but the one from the Cayley graph that looks like the permutohedron ( compare). The article says only permutohedron, not Cayley graph. Does anyone know an assignment of the weak orderings / ordered set partitions to the (hyper-)faces of this graph, where the assignment of the 24 permutations to the vertices is like in the permutohedron?
I know, that the flat representation of the permutohedron of order 4 is not actually a permutohedron, but I think that a drawing of the solid with 75 labels will be quite unusable - just like the drawing in the PDF, which does not even include all labels. mate2 code 23:40, 26 May 2013 (UTC)
The article is quite incomprehensible, especially as it doesn't succeed in defining what a weak ordering is. Madyno ( talk) 21:47, 28 May 2017 (UTC)
I removed the claim from the intro that posets are generalizations of weak orders, pointing out that not every weak order is a poset. The edit was reverted with the comment "nevertheless, the information contained in a weak order is a special case of the information contained in a partial order".
I do not agree with this statement. Partial orders can express the fact that certain elements are incomparable, while weak orders can express the fact that certain elements are tied in rank. Neither can express the other type of information, and neither type of information is a special case of the other.
For example, the set of physical quantities is partially ordered by size: 1 meter is less than 1 mile and 1 pound is less than 2 pounds, but 2 pounds and 1 meter are incomparable (and not tied in rank!). The set of colors is weak ordered by my preferences: I prefer red over green but I like green and blue equally (though they are comparable!).
I would mention in the intro that weak orders, just like partial orders, generalize total orders and are generalized by preorders. Total orders are precisely those relations that are both partial orders and weak orders. AxelBoldt ( talk) 03:09, 29 October 2020 (UTC)
Would you say that therefore rings and topological spaces generalize groups?Realizing a weak order as a poset involves no extra structure, so these are not very good analogies; a better analogy would be that monoids generalize groups. But, even so, the answer to your question is "yes". In fact, this perspective is very fruitful, at least with respect to rings: it motivates the representation theory of algebras from the representation theory of groups via the group algebra. -- JBL ( talk) 17:53, 29 October 2020 (UTC)
I changed this
If is incomparable with then for all satisfying either () or () or ( is incomparable with and is incomparable with ).
To this
If is incomparable with then for all , either () or () or ( is incomparable with and is incomparable with ).
And David Eppstein reverted this with justification "None of the three consequences is true for z=x"
I believe that If z=x then z is incomparable with x by irreflexivity and incomparable with y by substitution and the assumption that x is incomparable with y, so the third consequence is true. Anomalistic ( talk) 19:26, 23 January 2023 (UTC)
This article is rated C-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
However, not for every strict weak order there is a corresponding real function. For example, there is no such function for the lexicographic order on Rn.
But is seems easy to derive that there is such a function for Qn × R. We can add that if there is a source.-- Patrick 11:31, 21 May 2007 (UTC)
…and weak order gets redirected here? How is this different from making strict partial order a main article that contains a definition of partial order as a variant of the primary topic? I propose turning things around (for weakness, not for partiality).— PaulTanenbaum ( talk) 03:21, 18 March 2008 (UTC)
Currently there's a pretty picture on the page (like a snowflake or poinsettia) that does not illustrate what a strict weak order is. (It illustrates something else, which is not entirely irrelevant; I'm not saying the picture is misleading, just that it's not the one most important picture we need to have on this page.)
It would be awesome if we could just show the graph of a strict weak order on some smallish set. Strict weak orders are really very easy to understand; I think this is a topic where a good picture would be worth all the rest of the words on the page. — Jorend ( talk) 04:31, 13 December 2009 (UTC)
I've made two illustrations, that are inspired by drawings in this PDF (pages 23 and 26):
The assignment of the permutations to the vertices is not the one from the permutohedron ( compare), but the one from the Cayley graph that looks like the permutohedron ( compare). The article says only permutohedron, not Cayley graph. Does anyone know an assignment of the weak orderings / ordered set partitions to the (hyper-)faces of this graph, where the assignment of the 24 permutations to the vertices is like in the permutohedron?
I know, that the flat representation of the permutohedron of order 4 is not actually a permutohedron, but I think that a drawing of the solid with 75 labels will be quite unusable - just like the drawing in the PDF, which does not even include all labels. mate2 code 23:40, 26 May 2013 (UTC)
The article is quite incomprehensible, especially as it doesn't succeed in defining what a weak ordering is. Madyno ( talk) 21:47, 28 May 2017 (UTC)
I removed the claim from the intro that posets are generalizations of weak orders, pointing out that not every weak order is a poset. The edit was reverted with the comment "nevertheless, the information contained in a weak order is a special case of the information contained in a partial order".
I do not agree with this statement. Partial orders can express the fact that certain elements are incomparable, while weak orders can express the fact that certain elements are tied in rank. Neither can express the other type of information, and neither type of information is a special case of the other.
For example, the set of physical quantities is partially ordered by size: 1 meter is less than 1 mile and 1 pound is less than 2 pounds, but 2 pounds and 1 meter are incomparable (and not tied in rank!). The set of colors is weak ordered by my preferences: I prefer red over green but I like green and blue equally (though they are comparable!).
I would mention in the intro that weak orders, just like partial orders, generalize total orders and are generalized by preorders. Total orders are precisely those relations that are both partial orders and weak orders. AxelBoldt ( talk) 03:09, 29 October 2020 (UTC)
Would you say that therefore rings and topological spaces generalize groups?Realizing a weak order as a poset involves no extra structure, so these are not very good analogies; a better analogy would be that monoids generalize groups. But, even so, the answer to your question is "yes". In fact, this perspective is very fruitful, at least with respect to rings: it motivates the representation theory of algebras from the representation theory of groups via the group algebra. -- JBL ( talk) 17:53, 29 October 2020 (UTC)
I changed this
If is incomparable with then for all satisfying either () or () or ( is incomparable with and is incomparable with ).
To this
If is incomparable with then for all , either () or () or ( is incomparable with and is incomparable with ).
And David Eppstein reverted this with justification "None of the three consequences is true for z=x"
I believe that If z=x then z is incomparable with x by irreflexivity and incomparable with y by substitution and the assumption that x is incomparable with y, so the third consequence is true. Anomalistic ( talk) 19:26, 23 January 2023 (UTC)