Characterizations of finitary and cofinitary binary matroids (Q1117941)

This paper extends the many different characterizations of finite binary matroids to finitary and cofinitary matroids on infinite sets. In addition to the classical condition and by generalization of Lucas' results, we obtain the following. Let M(S) be a finitary or a confinitary matroid and for a fixed base B of M(S), let us denote by (C(x): \(x\in S-B)\) the family of fundamental circuits defined by B. The following statements about M(S) are equivalent: (1) M(S) is binary. (2) For every non-empty set \(\{x_ 1,...,x_ n\}\subseteq S-B\), the symmetric difference \(C(x_ 1)\Delta...\Delta C(x_ n)\) contains a circuit. (3) (i) M(S) has no minor isomorphic to the matroid \(W^ 3\) (whirl of order 3), and (ii) for x, \(y\in S-B\), \(x\neq y\), C(x)\(\cap C(y)\neq \emptyset\) implies C(x)\(\Delta\) C(y) is a circuit. (4) For every modular pair \((C_ 1,C_ 2)\) of circuits, \(C_ 1-B\neq C_ 2-B\).