A finite matroid $M$ is a pair $(E,I)$ where $E$ is a finite set and $I$ is a family of independent set with the following properties:


1) There is at least an independent system


2) Every subset of an independent set is independent.


3) If $A$ and $B$ are two independent sets and $A$ has more elements than 
$B$. Then, there exist an element in $A$ that when we added to $B$ gives a larger independent set than $B$.


The first two properties define a combinatorial structure known as an independence system.


1. Suppose that $M$ is an independence system but not a matroid. I am wondering if there is there a canonical... or a better... or unique way to construct a matroid from $M$ ?
2. Can we express $M$ canonically in terms of given matroids? (for example, as an intersection...)


Disclaimer: This question was posted first in math stack exchange without success...



matroids axioms and independence system
The substantial question 2 being already answered, I will address the less substantial 1.


Even if not substantial, the formalization of 1 might not be immediate (and at the end all the question might be considered by someone only a logical sophism, but since questions of the kind "is a canonical construction available?" arise naturally,I will answer anyway. In some cases the answer to such questions depends upon subtle corners of the formalization).


First note the following formal answer: yes, take as independent every (nonempty) subset of $E$.


It answers the question as posed (it is even the "largest" matroid extending $M$ on the same base set $E$),but it is clearly an undesired answer. It is a matroid generally too far from the initial independence system.


So let us search a "minimal" matroid extending a given independence system (minimal for set inclusion; by the substantial answer to 2 the independence system is the intersection of such matroids).


If the base set $E$ is not only a mere set, but a "list" (a finite set equipped with a total order), then again an (again unwanted, trivial) answer is available: take the first minimal matroid that extends $M$ (the first for the the "natural" order on $P(P(E))$: note that if $X$ is a totally ordered finite set, then so is also $P(X)=2^X$ with the lexicographic order).


This is again (formally correct but) unwanted since (a) no intelligent construction of the matroid is given (only a brutal "try each axiom and see"); (b) the answer depends on a (unrelated with $M$) total list order on $E$.


So one might request a request at least independence from the listing order.


Really, in general, in questions like this (can I "canonically construct" a structure of type $A$ on $E$ starting from a given structure of type $B$?) one requests "functoriality" (concrete over the base set $E$) at least for isomorphisms; in particoular, the group of transformations of $E$ that are $B$-automorphisms must also be $A$-automorphisms. 


[This must be so for any kind of reasonably first order definable constructions; see Hodges, model theory. In this specific case, the structures are not directly given as first order ones, but one can cripto-equivalently define them as first order structures with a standard trick: in place of $E$, take a atomic boolean algebra (so that $E$ is identified with its atoms); a added unary predicate "independent" applied to elements of the boolean algebra permits first order translations of the axioms .(using in place of 3 the form of the exchange axiom that works also for infinite dimensional matroid lattices: the covering property)]


With this "functoriality" request to formalize "canonical", an easy example shows that in general a canonical construction is impossible:


on a set $E$ with three elements $a,b,c$ take as maximal independent sets the singleton of $a$ and the pair of $b,c$. To minimally extend to a matroid one has to take as basis besides $b,c$ also either $c,a$ or $a,b$. 


(think $b,c$ as a horizontal and a vertical vector in the plane. As $c$ one takes either another horizontal or another vertical vector; non minimal matroid extensions take as $c$ a oblique vector in the plane or a vector outside the plane).


The group of automorphisms of the initial independence system contain the permutation that exchanges $b,c$ (and no other non-identity element). This exchanges the two minimal completion instead of being an automorphism for one of them, so the impossibility of "functoriality" is obtained.


