Alternative definition of and opposite concept to a matroid?

Question

From Wikipedia

In terms of independence, a finite matroid $M$ is a pair $(E,\mathcal{I})$, where $E$ is a finite set (called the ground set) and $\mathcal{I}$ is a family of subsets of $E$ (called the independent sets) with the following properties:

• The empty set is independent, i.e., $\emptyset\in\mathcal{I}$. Alternatively, at least one subset of $E$ is independent, i.e., $\mathcal{I}\neq\emptyset$.

• Every subset of an independent set is independent, i.e., for each $A'\subset A\subset E$, if $A\in\mathcal{I}$ then $A'\in\mathcal{I}$. This is sometimes called the hereditary property.

• If $A$ and $B$ are two independent sets of $\mathcal{I}$ and $A$ has more elements than $B$, then there exists an element in $A$ that when added to $B$ gives a larger independent set. This is sometimes called the augmentation property or the independent set exchange property.

A subset of the ground set E that is not independent is called dependent. A maximal independent set—that is, an independent set which becomes dependent on adding any element of E—is called a basis for the matroid.

1. Can a matroid be defined equivalently by replacing the augmentation property with the following one :

   • $\forall A \in \mathcal{I}$, $A$ has a maximal superset in $\mathcal{I}$, and the cardinality of the maximal superset of $A$ is the same for all members of $\mathcal{I}$.

2. I was wondering if there has already existed a concept which is opposite to a matroid? For example, for a pair $(E, \mathcal{J})$,

   • $E \in \mathcal{J}$,

   • for each $A'\subset A\subset E$, if $A' \in\mathcal{J}$ then $A \in\mathcal{J}$.

   • If $A$ and $B$ are both in $\mathcal{J}$ and $A$ has less elements than $B$, then there exists an element in $B$ when removed from $B$ gives a smaller member in $\mathcal{J}$. ($A$ may or may not be helpful in finding such an element in $B$.)

   or can the third point be replaced by

   • $\forall A \in \mathcal{J}$, $A$ has a minimal subset in $\mathcal{J}$, and the cardinality of the minimal subset of $A$ is the same for all members of $\mathcal{J}$.

   Based on the definition, can we further define a concept just opposite to a basis of a matroid, something like "a minimal set in $\mathcal{J}$ is called a basis for $(E, \mathcal{J})$".

   An example of such $(E, \mathcal{J})$ will be the collection of all bases of a topology.

Thanks and regards!

Answer 1

For the first question, the augmentation property cannot be replaced by this new statement. This statement essentially says that all maximal independent sets (bases) are of same size. But this is not sufficient to prove the augmentation property. Consider this example, with $E = \{ 1,2,3,4\}$ and $I = \{ \phi, \{1\}, \{2\}, \{1,2\}, \{3\}, \{4\}, \{3,4\}\}$. This satisfies your new statement, but not the augmentation property. When trying to describe in terms of bases, a property similar to augmentation is used. In terms of independent sets, another typically used definition is to replace augmentation by the following property.

• Given any $A \subseteq E$, all maximal independent sets contained in $A$ have the same size.

Answer 2

As @polkjh points out, the augmentation property is not equivalent to the property that all maximally independent sets have the same size. While (the independent sets of) any matroid form a pure simplicial complex, not every pure simplicial complex is a matroid. The example given in @polkjh's answer of the simplicial complex on $\{1,2,3,4\}$ with facets $\{1,2\},\{3,4\}$ is a minimal example.

Now let's look at the second collection of axioms. The first three taken together imply that the resulting set system $\mathcal{J}$ always consists of all subsets of $E$. To see this note that $E \in \mathcal{J}$ by the first axiom. Repeatedly applying the third axiom we find that the empty set is also in $\mathcal{J}$. Now apply the second axiom to see that all subsets of $E$ are in $\mathcal{J}$.

Note that a set system satisfying the third axiom is called an accessible set system. It is used in the definition of a greedoid, a type of set system generalizing both matroids and antimatroids.

Replacing the third axiom by the fourth yields set systems that are precisely the nonfaces of a simplicial complex whose Stanley-Reisner ideal is generated in a single degree. Such a simplicial complex need not be pure: for example the Stanley-Reisner ideal of the simplicial complex on $\{1,2,3,4\}$ with facets $\{1,2,4\},\{1,3,4\},\{2,3\}$ is generated by the monomials $x_1x_2x_3$ and $x_2x_3x_4$. Here's a picture to help visualize the situation. The facets are shaded blue while the smaller faces are circled in blue. Similarly the minimal nonfaces are shaded gray while the non-minimal nonface is circled in gray.

Answer 3

My favorite "alternative" definition of a matroid is: A matroid is a simplicial complex whose every induced subcomplex is pure.