Statement of Research

Matroid theory is a generalization of the idea of linear independence. A matroid M consists of a finite set E (called the ground set) and a collection S of subsets of E satsifying the following conditions: (1) ∅ ∈ S; (2) if I ∈ S, then every subset of I is in S; (3) if I1 and I2 are in S and |I1| < |I2|, then there is an element e of I2 − I1 such that I1 ∪ e is in S. The elements of S are called independent sets. For example, given any matrix, let its collection of column vectors be the the ground set E of a matroid. A set A of columns is independent whenever it forms a linearly independent set in the sense of linear algebra. Then the collection of independent sets satisfies the conditions of a matroid. Every matrix thus represents a matroid, but not every matroid can be represented by a matrix. Those that can make up a special class of matroids called the representable matroids. Some matroids are representable over certain fields but not others. Similarly, graph theory is generalized by matroid theory. For any graph, the edge set can play the role of the ground set in the above definition. An independent set is then a collection of edges which contains no cycles. Again, we have satisfied the conditions of a matroid. Not every matroid can be represented by a graph. Those that can are called graphic matroids (note that all graphic matroids are representable). My results hold for all matroids. We will continue to use graphic examples to inform our intuition, but all of the constructions that we will discuss extend to matroids in general. Once we have established our collection of independent sets, we can define the rank r(A) of a set A to be the size of its largest independent subset. The closure cl(A) is A together with all elements e ∈ E such that r(A ∪ e) = r(A). A is called a closed set if cl(A) = A. The closed sets (or flats) of a matroid form a geometric lattice. In fact, there is a one-to-one correspondence between simple matroids (those with no loops or multiple edges) and geometric lattices. My research focuses on geometric lattices. In particular, I have looked at a certain type of operation—a join congruence relation—on geometric lattices. A join congruence relation is an equivalence relation on the elements of a lattice with the additional property that x ≡ y implies x∨ a ≡ y ∨ a. Given a join congruence relation θ on a lattice L, we consider the set of equivalence classes of θ. The collection of equivalence classes forms a lattice [4], which we denote L/θ.