Talk:Matroid

message: From the article:
:"a matroid or independence structure "
This is NOT the same. An independance structure does NOT have to fulfill the third independance axion (according to "Korte/Vygen - Combinatorial Optimization" and the german wikipedia article). 84.60.98.171 (talk) 16:10, 8 February 2014 (UTC)
thread_title:  Independence structure 

message: I have a question about this paragraph:
::The first two properties are simple, but the motivation behind the third property is not obvious. The property implies the fact that given two independent sets of the same size, any element of one can be replaced by some element of the other; thus the name "exchange." The examples in the example section below will make its motivation clearer.
I will try to construct an counter-example using the tree/forest/graph example:
Imagine 2 graphs with 5 vertices and 3 edges each:
x---x             x   x
    |     and       \ |
x   x             x   x
 \                   /
  x                 x
Like this we should have two independent sets of the same size.
Now the above paragraph says: We can replace any elemnt between the two forests and they should stay indepent (means circular-free).
But what about this exachange though:
x---x             x   x
  \ |     and         |
x   x             x   x  
                   \ /
  x                 x
So now there is a circle in the left forest --> we don't have two independent sets of the same size anymore, or???
And like this the above paragraph should be wrong?!
Any hint / help / correction would be very apreciated!
Tormen 11:36, 27 January 2006 (UTC)
replies: text: Right you are. An excellent counterexample. I have no idea what I was thinking when I wrote that - I guess I have no idea why it's called the exchange property after all. A better name would be like "the expansion property" or something. Deco 17:24, 27 January 2006 (UTC)

text: That is still not correct.  Elements are not swapped.  The exchange property says there is an element in the second such that, putting it into the first set and removing the original first-set element, you get an independent set.  You do not expect anything nice to happen in the second set. Zaslav 02:02, 24 February 2007 (UTC)
replies: text: True.  My choice of the word swap was poor.  I was only referring to what you have said. Lmdemasi 14:34, 9 March 2007 (UTC)
thread_title:  Exchange property paragraph question 

message: the following nonexample removed as incorrect:
:On the other hand, consider this non-example: let E be the vertices of a graph, and let the independent sets be those that can be colored with three colors. The empty set can be three-colored, and any subset of a three-colorable set is three-colorable, but the exchange property does not hold, because it's possible to have two maximal three-colorable subgraphs of different sizes, as shown to the right.
The picture is wrong: the text speaks about colorable, but the pucture shows colored. The left graph is colorable in 3 colors (only in a different way), hence the (1,2,3) is not maximal colorable. `'mikka (t) 16:27, 30 April 2006 (UTC)
replies: text: Right you are. I wrote this example and rewrote it to be correct, speaking of colorings instead of colorability. Dcoetzee 01:27, 11 July 2007 (UTC)
thread_title:  non-example 

message: This article mentions "coatom" without defining what it is. Please fix. - 74.112.174.192 20:00, 25 February 2007 (UTC)
replies: text: Thank you.  It's done. Zaslav 09:12, 27 February 2007 (UTC)
thread_title: Definition of "coatom"

message: The text states:
:If E and F are disjoint, the union is the direct sum.
Should that be "the union is isomorphic to the direct sum" or some such?  The Wikipedia definition of the disjoint union of two sets does not reduce to the union when the sets are disjoint, and I assume that matroids over two different underlying sets cannot be the same matroid.  LachlanA 21:22, 10 July 2007 (UTC)
replies: text: The WP definition has two forms.  The appropriate definition for direct sum is a union in which the two sets are disjoint. Zaslav (talk) 06:18, 26 November 2012 (UTC)
thread_title:  Matroid union vs Direct sum 

message: I could be wrong about this, but I'm pretty sure the circuits of the Fano matroid are the sets of three points with curves through them and their complements.
The description given appears to imply that only the sets of three points on curves are circuits. It doesn't necessarily imply this, but it does appear to at first sight.
Can someone tell me if, as I think, the complements of these sets are also circuits, as if so I'd like to edit the page for clarity.
Thanks!
James mcl (talk) 16:30, 3 June 2008 (UTC)
replies: text: You're right.  Since the matroid has rank 3 and is simple, any set of 4 points that doesn't contain a circuit of 3 points (i.e., a 3-point line) is predictably a circuit.  Thus, one could say the 3-point lines are the "essential" circuits.  However, they aren't the only circuits and the article should not give that impression.  Zaslav (talk) 04:56, 4 July 2009 (UTC)
thread_title:  The Fano matroid 

message: === Takeo Nakasawa: The Forgotten Father of Matroid Theory ===
This book is about a japanese mathematician who worked on matroids and is not mentioned in the article:
http://www.amazon.com/gp/product/3764385723  —Preceding unsigned comment added by 201.53.116.195 (talk) 14:04, 12 May 2009 (UTC)
thread_title: Mathematicians deserving mention

message: This book is about a japanese mathematician who worked on matroids and is not mentioned in the article:
http://www.amazon.com/gp/product/3764385723  —Preceding unsigned comment added by 201.53.116.195 (talk) 14:04, 12 May 2009 (UTC)
thread_title:  Takeo Nakasawa: The Forgotten Father of Matroid Theory 

message: Birkhoff's paper and his lattice-theoretic work may deserve mention, also.  Kiefer.Wolfowitz 09:49, 15 April 2012 (UTC)
thread_title: Garrett Birkhoff

message: The section on "matroid software" seems irrelevant to the rest of the article.  It is not "encyclopedic".  
It contains many external links, and seems that the writer of this section is simply "plugging" his/her own projects. http://en.wikipedia.org/wiki/Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_a_directory Jwesley78 (talk) 04:45, 12 September 2009 (UTC)
replies: text: Plugging something, that's for sure!  It's very poor.  I think I can improve it quickly, at any rate. Zaslav (talk) 02:58, 13 September 2009 (UTC)
thread_title: Matroid Software

message: Hey, does anyone actually have citations where one could look up the two theorems and their proofs cited in the "Greedy Algorithms" sections? I've seen them (vaugely) attributed to Edmonds, but I'm having the hardest time actually finding them in his paper Bruce IV (talk) 00:32, 1 December 2011 (UTC)
replies: text: See Oxley's book.  It will help you.  Zaslav (talk) 06:13, 26 November 2012 (UTC)
replies: text: Pages 62-64 of the 1992 edition to be precise.  The assertion about the independence oracle is more-or-less explicit.  Deltahedron (talk) 22:20, 26 November 2012 (UTC)
thread_title:  Greedy Algorithms 

message: I am reading this topic having little prior knowledge of it.  It is nice that the exposition is very teacherly, but seems to contain a few confusing non-sequitors.  For instance, after giving the three properties of matroids it is claimed that the first two are obviously motivated but the third isn't.  Why??  This is familiar from linear algebra as the statement that a basis for a subspace can be extended to a basis of the ambient space.  
Just after this it is said concerning circuits, that "in spite of the similarity to the definition of basis this notion has no analogue in classical linear algebra."  The only "similarity" to the definition of a basis is that it is the opposite in a certain sense.  That is not much of a similarity and no one knowing linear algebra would think that a minimally dependent set is anything like a basis.  At the same time one could easily define such a thing --for instance in finite-dimensional linear algebra-- but it's not a useful concept there.
Geminatea (talk) 18:17, 18 March 2012 (UTC)
replies: text: I think your removal of this editorialization is an improvement — thanks. —David Eppstein (talk) 18:37, 18 March 2012 (UTC)

text: I agree completely. Zaslav (talk) 02:28, 15 April 2012 (UTC)
thread_title:  Expository Features 

message: The definition A matroid M is called a frame matroid if it, or a matroid that contains it, has a basis such that all the points of M are contained in the lines that join pairs of basis elements appears to be taken from .  On the other hand, there appears to be a different definition at   Are these in fact the same, and if so, is there a reliable source that reconciles them, or is this terminology non-standard?  Deltahedron (talk) 20:31, 30 November 2012 (UTC)
replies: text: The paper of Zaslavsky shows that the definitions are equivalent.  Zaslav (talk) 04:10, 2 December 2012 (UTC)
thread_title:  Frame matroid 

message: I presume that in the formula 
:p_M(\lambda) := \sum_A \mu(\emptyset,A) \lambda^{r(M)-r(S)} 
the otherwise undefined function μ is the Möbius function of the matroid lattice?  Is there a reliable source for this?  Deltahedron (talk) 22:19, 1 December 2012 (UTC)
replies: text: Did you add the WP link for μ?  If so, thank you.  Is the White citation for the characteristic polynomial?  If so, ditto.  Zaslav (talk) 04:28, 2 December 2012 (UTC)
thread_title:  Characteristic polynomial 

message: Everyone knows that Prof. Dr. Korte is a matroid-genius´, but why is his name not mentioned in this article.
Add at least his book: Korte, Vygen - Combinatorial Optimization  — Preceding unsigned comment added by 87.161.21.168 (talk) 16:29, 2 October 2013 (UTC)
replies: text: Bernhard Korte's Wikipedia article does not mention matroids, and my impression is that he is better known for his work on greedoids (although he has a reasonably well-cited paper with Jensen in SICOMP 1982 on computational complexity issues related to matroids).  In any case, what should his book be added as a reference for, that does not already have better references listed? —David Eppstein (talk) 17:54, 2 October 2013 (UTC)
thread_title: KORTE

message: The matroid page should be improved by adding the following information:
1) clearly define substructures (submatroids) and quotient structures,
2) clearly define morphisms of matroids, especially matroid isomorphisms,
3) more rigorously define matroid representability using matroid morphisms,
4) describe isomorphism classes for small cardinalities of the underlying set,
5) describe the current progress in matroid enumeration and classification of matroid isomorphism types.  — Preceding unsigned comment added by Peteris.daugulis (talk • contribs) 16:50, 28 December 2013 (UTC)
thread_title: Possible improvements

message: The point is supposed to be that the middle four points are *not* on a plane, so it's confusing that there is a plane drawn through them.  See for example figure 17(c) and its caption on page 28 of these notes by Vic Reiner. Ctourneur (talk) 03:44, 8 April 2015 (UTC)
replies: text: I'm not convinced. Reiner's description in the figure caption seems to contradict both this source (used in the Vámos matroid article) and Reiner's figure itself (what are edges ce and df there for, if not to show that cdef is a fifth coplanar 4-tuple?). Besides, a matroid with only the outer four 4-tuples coplanar is easy to realize in 3d and therefore linear, unlike the Vámos matroid. Perturb the six planes of a cube into general position (so that each pair of opposite edges becomes skew) and then choose four used-to-be-parallel edges (say the vertical ones) and perturb all eight vertices along the lines of those edges into general position (so that the top and bottom faces of the cube stop being coplanar). My guess is that the omission of the fifth 4-tuple from Reiner's notes is just a mistake. —David Eppstein (talk) 04:17, 8 April 2015 (UTC)
replies: text: Thanks, I think you're right.  The crucial thing I was missing is that the top line and the bottom line (as drawn in the Wikipedia picture) are supposed to be skew, which isn't super obvious from the picture.  Ctourneur (talk) 01:23, 9 April 2015 (UTC)
thread_title: Mistake in Vamos matroid picture

message: It seems like this page should mention minimum spanning trees at least once.  — Preceding unsigned comment added by 73.222.120.136 (talk) 02:58, 17 October 2015 (UTC)
thread_title: Minimum spanning trees

message: In the second axiom for independence it states:
for each A'\subset A\subset E, if A\in\mathcal{I} then A'\in\mathcal{I}
Shouldn't this be
for each A'\subset A\subseteq E, if A\in\mathcal{I} then A'\in\mathcal{I}
since A might be the entire ground set. Otherwise this test wouldn't apply to E=\left\{1, 2, 3\right\}, \mathcal{I}=\left\{\left\{1, 2, 3\right\}\right\} since \left\{1, 2, 3\right\} isn't a strict subset of \left\{1, 2, 3\right\}  — Preceding unsigned comment added by Lee A. Christie (talk • contribs) 10:58, 19 November 2015 (UTC)
thread_title:  Axiom I2 Subset E 

message: In axiom/definition 3,  "A has more elements than B" seems a poor/loose definition if not pointing to the related theory.
Can it be clarified? (from which theory it is taken, axiomatic set theory? vector space ...)  — Preceding unsigned comment added by 87.67.113.207 (talk) 06:25, 19 December 2015 (UTC)
replies: text: You need to specify an axiom systems to be able to count? Everything is finite here, so the number of elements is just a natural number. —David Eppstein (talk) 06:35, 19 December 2015 (UTC)
thread_title:  Definition 

message: What if we merely omit finiteness from the definition? Same question about definition of matroid from vector space, at least finite-dimentional.79.179.15.250 (talk) 13:31, 17 April 2017 (UTC)
replies: text: infinite matroids are subject of current research e.g.  most of classical definitions e.g. Whitney and Tutte are all for the finite case. Going to the infinite case is quite a big step, in layman terms is like the difference between a finite polynomial and an analytical function or between algebraic and non algebraic. Historically in Quantum Mechanics there are a lot of references around infinite matrixes but from a rigorous mathematical standpoint they are not well accepted, in fact it becomes about operators and Hilbert spaces. In the scope of matroids you can imagine that a functional definition/representation of matroids like rank functions is similar/analogous to functors or operators on hilbert spaces, where for example an infinite basis can be well defined. 
Finally matroids belong to the realm of non-pure mathematics e.g. see the reference of Pierre Cartier in the Nicolas_Bourbaki page which means that sometime there is often little integration from something like matroids that is assigned to belong to "applied mathematics" with the "pure math stuff". Matroids have huge relationships with algebra and topology and this "infinite matroids" is actually a topic that bridges from matroids to the other two.  — Preceding unsigned comment added by Flyredeagle (talk • contribs) 10:13, 5 November 2020 (UTC)
thread_title:  Why finite? 

message: Missing a section on applications - whether theoretical or practical PolyCreator (talk) 21:17, 16 May 2022 (UTC)
thread_title:  Applications? 

message: The article currently says:
"If E is any finite subset of a vector space V, then we can define a matroid M on E by taking the independent sets of M to be the linearly independent subsets of E. [...]  If M is a matroid that can be defined in this way, we say the set E represents M.  Matroids of this kind are called vector matroids. [...] A matroid that is equivalent to a vector matroid, although it may be presented differently, is called representable or linear.  If M is equivalent to a vector matroid over a field F, then we say M is representable over  in particular, M is real-representable if it is representable over the real numbers.  For instance, although a graphic matroid (see below) is presented in terms of a graph, it is also representable by vectors over any field."
It's not clear to me which distinction is being made here between vector matroids and representable/linear matroids. The term "equivalent" hasn't been defined (this is a problem in itself); I presume it's meant to be synonymous with "isomorphic" (which also hasn't been defined but which is generally understood to refer to the existence of a structure-preserving bijection). But if two matroids M and M' are isomorphic and M can be defined as a finite subset of a vector space (with independent sets being the linearly independent sets), then M' can also be defined in this way (simply by considering the same vector space with the elements of the M replaced by those of M'). (The example of a graphic matroid even explicitly states that a graphic matroid "is also representable by vectors".) But then "vector matroid" is synonymous with "representable/linear matroid".
I suspect that what the article might be trying to say is that "vector matroid" is an informal term that one uses for matroids that are defined via a vector space construction, whereas "representable/linear matroid" is a formal term that refers to matroids that can be constructed in this way, whether they naturally arise in this way or are defined via another construction and just turn out to also be constructible in this way. But that's not at all obvious from the current formulation – by saying that "Matroids of this kind are called vector matroids", where "matroids of this kind" refers to "a matroid that can be defined in this way", "vector matroid" is being defined to mean the same thing as "representable/linear matroid".
Or am I missing something?
Joriki (talk) 00:18, 17 February 2023 (UTC)
thread_title:  vector matroid vs. representable or linear matroid 

message: Examples/Matroids from graph theory:
"Every finite graph (or multigraph) G gives rise to a matroid M(G) as follows: take as E the set of all edges in G and consider a set of edges independent if and only if it is a forest; that is, if it does not contain a simple cycle." 
Now in https://en.wikipedia.org/wiki/Cycle_(graph_theory), you can read:
"A cycle or simple circuit is a circuit in which only the first and last vertices are equal."
Thus, according to Wikipedia, there is no such notion as simple cycle. There is just a simple circuit, and that is a cycle. 2001:4C4D:1C08:8100:EDF4:7965:F231:D213 (talk) 11:19, 29 December 2023 (UTC)
replies: text: In the same article you can also read "The term cycle may also refer to an element of the cycle space of a graph." The adjective "simple" disambiguates this as referring to the first meaning, not the second. The phrase "simple cycle" is very common on the mathematical literature. You should not read the Wikipedia cycle article as forbidding its use. —David Eppstein (talk) 15:43, 29 December 2023 (UTC)
thread_title:  simple cycle ?!