Equivalent definitions for a coloop?

Question

1. From wikipedia, in a matroid,

   An element that belongs to no circuit is called a coloop. Equivalently, an element is a coloop if it belongs to every basis.

   I wonder why the equivalence?

   From the equivalence, I suspect that the union of all circuits and the intersection of all bases do not overlap? Is it correct?

2. I also saw that a coloop in a matroid can be defined as a loop in its dual matroid. Why is this equivalent to the previous two definitions?

   I understand that A loop is an element which is also a circuit. Equivalently, it lies in no basis.

   The complement of a circuit in a matroid is a hyperplane in its dual matroid. In its dual matroid, the complement of the hyperplane is a coloop. So in a matroid, is the complement of a hyperplane inside every basis?

Thanks.

Answer 1

An element that belongs to no circuit is called a coloop. Equivalently, an element is a coloop if it belongs to every basis.

Circuits are minimal dependent sets, while bases are maximal independent sets.

Now let $c\in E$ be a coloop, so it belongs to no circuit. Let $B\subseteq E$ be a basis and assume $c\notin B$. Since $B$ is maximal independent, $B\cup\{c\}$ is dependent, so it contains a circuit $C\subseteq B\cup\{c\}$. Now $C\subsetneq B$, since $C$ would be independent otherwise. So we conclude $c\in C$, which contradicts $c$ being a coloop. Thus, our assumption was wrong and we have $c\in B$, so $c$ is an element of every basis.

I leave it to you to find a proof for the other implication.

From the equivalence, I suspect that the union of all circuits and the intersection of all bases do not overlap? Is it correct?

Indeed, the union of all circuits does not intersect the intersection of all bases, since the latter is the set of coloops, while the former doesn't contain any coloop.

I also saw that a coloop in a matroid can be defined as a loop in its dual matroid. Why is this equivalent to the previous two definitions?

For the dual matroid, the bases are exactly the complements of the original bases. So something that is a loop in the dual matroid belongs to the complement of no basis of the matroid itself, thus is contained in every basis, so it is a coloop.

The complement of a circuit in a matroid is a hyperplane in its dual matroid. In its dual matroid, the complement of the hyperplane is a coloop. So in a matroid, is the complement of a hyperplane inside every basis?

A hyperplane is a flat of rank $r-1$, where $r$ is the rank of the matroid. Thus, adding any element to a hyperplane gives you a set of full rank, so it contains a basis. Note that a hyperplane doesn't always have $|E|-1$ elements, so your second statement can't be true in general, since the coloops are elements.

Answer 2

It might be helpful to consider graphic matroids. Here coloops correspond to bridges. We have the statement that coloops are not in any circuit which is equivalent to coloops being in every base. This corresponds to the statement that bridges are not in any cycle which is equivalent to bridges being in every spanning tree (forest).

Now circuits are minimally dependent sets while bases are maximally independent sets. So, the point is that adding a coloop to an independent set will always give a larger independent set and never will never turn an independent set into a dependent set. This can also be formulated using matroid rank functions. That is adding a coloop will always increase the rank of a set.

Again the fact that loops and coloops are dual can be see that in the graphic case where loops and bridges are (planar) dual to each other.

To explicitly answer some of your questions. Yes, the intersection of bases (coloops) and unions of circuits do not overlap. I am not familiar with the notion of a hyper plane inside a matroid.

In the general setting we get the dual matroid by complementing bases. Thus a coloop is in every base of the matroid and hence no base of the dual matroid. Since independent sets are just subsets of bases we see the dual of a coloop is a loop which is in no independent set. That is a loop which is dual to coloop is a dependent set on its own as a singleton. In terms of the rank function the set containing just a loop has rank zero.

Yes, the intersection of bases (coloops) and the union of circuits does not overlap.

I am unfamiliar with the use of hyperplane inside an arbitrary matroid.