The other part of my talk to the codgers was about a property of vector spaces which is far from true for matroids: the symmetry. Any map from a basis into the vector space extends uniquely to an endomorphism of the space.
Of course, endomorphisms have to be endomorphisms of something, so the right context is universal algebra. The definitions had already been made by semigroup theorists such as John Fountain and Victoria Gould: an independence algebra is an algebra (in the sense of universal algebra) satisfying the following conditions:
- the minimal generating sets satisfy the exchange property, so they are the bases of a matroid;
- any map from a basis into the algebra extends (uniquely) to an endomorphism of the algebra.
All that really matters in this definition is the lattice of subalgebras and the monoid of endomorphisms. So we say that two independence algebras on the same set are equivalent if they have the same subalgebras and endomorphisms; the task is to classify them up to equivalence.
Now the automorphism group of an independence algebra has the property that the pointwise stabiliser of any set fixes pointwise the flat it spans. So in particular the automorphism group is an IBIS group, as described in the preceding post. It has the additional property that it permutes (ordered) bases transitively.
Csaba Szabó and I were able to classify the finite independence algebras, and develop some theory for the general finite-rank case. Here is a very brief summary.
- A rank 1 independence algebra determines, and is determined by, a group G acting on a (possibly empty) set C; the underlying set is G∪C, where elements of C have rank 0 and those of G have rank 1. (This is a remarkably concise description of a group acting on a set!)
- Call an independence algebra trivial if its subalgebra lattice is isomorphic to the Boolean algebra of subsets of a finite set X. A trivial independence algebra is specified by a group G acting on a set C as above, together with the set X; in particular, the underlying set is (X×G)∪C.
- The subalgebra lattice of a non-trivial independence algebra is isomorphic to the subspace lattice of a projective or an affine space, depending whether the set of rank 0 elements is empty or not. The proof involves first showing that the rank–nullity formula for endomorphisms of a vector space holds for non-trivial independence algebras with rank 0 elements.
- Using Tracey Maund’s classification of the IBIS groups which permute their bases transitively (itself a consequence of the Classification of Finite Simple Groups), together with some detailed analysis of each type, one finds that the finite non-trivial independence algebras are of one of the following types:
- a vector space with a distinguished subspace of constants;
- an affine space with a distinguished subspace of translations;
- an affine version of a nearfield (a structure like a field but possibly lacking the commutative law and one distributive law).
Can this classification be given without using CFSG? (Note that Boris Zilber gave a version of Maund’s theorem for matroids of rank at least 7 without CFSG, though the proof is hardly simple!)