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Cryptomorphisms ∗ MAA Short Course Lecture Notes Jenny McNulty Department of Mathematical Sciences, The University of Montana Missoula, MT 59812-1032, USA [email protected] January, 2011 1 Introduction One of the most attractive features of matroids is the plethora of equivalent definitions. Borrowing from linear algebra, we can define a matroid in terms of independent sets or equivalently in terms of bases. Graph theory motivates alternate formulations of matroids in terms of circuits and cocircuits, while open and closed sets, closure and hyperplanes come from geometry. We will tie these disparate concepts together using the rank function. There are many additional ways to define a matroid; we'll discuss some of them here, but leave the rest for you to explore. A central idea of this topic is that we can convert from one axiom system to another. We'll describe a method, called a cryptomorphism, for converting from one system to an- other. A cryptomorphism is an \equivalent way to define matroids, yet not straight- forward" 1 While this variety of axiom systems (that is to say the number of cryptomorphic definitions a matroid) gives matroid theory much of its power, it can often contribute to the difficulty in learning the subject. We suggest the reader keep the prototypes in mind while learning the formal definitions and refer often to the cryptomorphism charts. 2 From Independent Sets to Bases and Back Again We begin with the formal definition of a matroid. Independent Definition 1. A matroid M is a pair (E; I) in which E is a finite set and I is a family of sets subsets of E satisfying (I1) I 6= ;; Non- triviality (I2) If J 2 I and I ⊆ J, then I 2 I; Closed un- (I3) If I;J 2 I with jIj < jJj; then there is some element x 2 J − I with I [ fxg 2 I: der subsets Augmentation ∗These notes are adapted from the text \Matroid Theory" by G. Gordon and J. McNulty. 1G. Birkhoff coined the term crypto-isomorphism in 1967 [1], since then the term has been shortened to cryptomorphism. 1 The family I forms the independent sets of the matroid. The three properties (I1), (I2), and (I3) listed above are not the only properties the independent sets of a matroid satisfy. For instance, it's easy to see that properties (I1) and (I2) together imply ; 2 I. The next proposition points out that one can define a matroid in many ways - even just restricting oneself to independent sets. Proposition 2. Let E be a finite set and let I be a family of subsets of E. Then the family I forms the independent sets of a matroid iff: (I10) ; 2 I; (I2) If J 2 I and I ⊆ J, then I 2 I; (I30) If I;J 2 I with jJj = jIj + 1; then there is some element x 2 J − I with I [ fxg 2 I: Definition 3. If M is a matroid with independent sets I, then B is a basis of the matroid M if B is a maximal independent set. Bases Let's see how we can define a matroid directly in terms of bases. Here are 3 things that are always true for the bases B of a matroid: (B1) B= 6 ;. Non- triviality (B2) If B ;B 2 B and B ⊆ B , then B = B : 1 2 1 2 1 2 Clutter (B3) If B1;B2 2 B and x 2 B1 − B2, then there is an element y 2 B2 − B1 so that Basis Ex- B1 − x [ fyg 2 B: change These properties that the collection B of bases satisfy is actually all we need to define a matroid in terms of bases. That is, these properties can be taken as an equivalent axiom system for a matroid. Just as the axiom system for independent sets mirrors the properties of linear independence in vectors, the axiom system for bases is motivated by properties satisfied by the collection of bases of a (finite-dimensional) vector space. Here's an alternate version of (B2): 0 (B2 ) If B1;B2 2 B, then jB1j = jB2j Equicardinal The next proposition, will be useful when we work through the details of the crypto- morphism between independent sets and bases. Proposition 4. Let E be a finite set and let B be a family of subsets of E. The family B satisfies (B1), (B2), (B3) if and only if B satisfies (B1), (B20), (B3). We are finally ready to discuss our first cryptomorphism. This involves the following steps: 1. Given the independent sets I in a matroid, define a collection of subsets B to be the maximal independent sets; 2. Next, prove that the collection B satisfies the properties (B1), (B20), and (B3); 3. Reverse the procedure: Given the collection of bases B of a matroid, define a collection of subsets I to be all subsets of the bases in B; 2 4. Now, prove the collection I you just defined satisfies the properties (I1), (I2), and (I3) (or, equivalently, (I10), (I2), and (I30)). We're still not done. When we define I in terms of B and also define B in terms of I, we are actually dealing with functions on the set of all families of subsets of E. Let f and g be these cryptomorphic functions with f(I) = B and g(B) = I (so f; g : 22E ! 22E ). We have to show these two cryptomorphisms compose correctly: 5. Show f(g(B)) = B, and now g(f(I)) = I. Maximal Bases . Independent Sets Subsets . Figure 1: Cryptomorphism between Independent Sets and Bases Theorem 5. Let E be a finite set and let B be a family of subsets of E satisfying (B1), (B2), (B3). Then, (E; B) is cryptomorphic to the matroid M = (E; I) and B is the collection of bases of a matroid. Example 6. Let M be the matroid on the six element set E = fa; b; c; d; e; fg with bases B = face; ade; bce; bde; cdeg. e f a b c d Figure 2: The matroid from Example 6. Some curious features of this matroid are the behavior of the elements e and f. Note that e is in every basis of the matroid. Equivalently, it can be added to any independent set it's not in to create another independent set. Such an element is called an isthmus or coloop of the matroid. On the other hand, the element f is in no basis { in fact, f is in no independent set at all. This element is called a loop. Both isthmus and loop are terms borrowed from graph theory. A set which is not independent is called dependent and a minimal dependent set is called a circuit. In the example, we list the collection C of all circuits: C = ff; ab; acd; bcdg. Listing the circuits is often an efficient way to describe a matroid. Definition 7. Let M be a matroid on the ground set E. An isthmus is an element x 2 E that is in every basis. A loop is an element x 2 E that is not in any basis. Isthmus Loop In summary, if we are given a matroid M = (E; I) defined in terms of independent we sets, can easily find the bases, the dependent sets and the circuits. Moreover, if we are given any one of these collections we can find the others { see Figure 3. For example we can define circuits C in terms of independent sets I and vice versa as: 3 •I!C: The circuits C are the minimal subsets of E that are not independent. •C!I: The independent sets I are all the subsets of E that don't contain a circuit C 2 C. The three key properties the family C satisfies are: (C1) ; 62 C; Non- triviality (C2) If C ;C 2 C and C ⊆ C , then C = C , 1 2 1 2 1 2 Clutter (C3) If C1;C2 2 C with C1 =6 C2; and x 2 C1 \ C2, then C3 ⊆ C1 [ C2 − x for some C3 2 C: Circuit Elimination Theorem 8. Let E be a finite set and let C be a family of subsets of E satisfying (C1), (C2), (C3). Then, (E; C) is cryptomorphic to the matroid M = (E; I) and C is the collection of circuits of M. Maximal Bases Independent Sets Subsets Not Not . Minimal . Dependent Sets Circuits . Supersets Figure 3: Cryptomorphism between Independent Sets, Bases, Dependent Sets and Circuits 3 Rank, Flats, Hyperplanes and Closure In this section, we define several additional matroid concepts having a geometric flavor. For each new term we define, we'll relate it to previously defined matroid concepts and describe the new term for geometries and graphs. As before, we can define a matroid directly in terms of these new concepts. The example we will use throughout this section is the matroid M on the six element set E = fa; b; c; d; e; fg with bases B = face; ade; bce; bde; cdeg { this is the same matroid we met in Example 6. In Figure 4, we draw the geometry corresponding to M and we also give a graphic representation. Example 9. In the picture of the matroid M on the left of Figure 4, the bases of M correspond to the collections of three points that are not collinear. Note that the isthmus e is in every such set. For the graph G on the right of Figure 4, note the bases correspond to the spanning trees of G, and this time, the edge corresponding to the isthmus e is an isthmus of G.
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