By Dawkins P.

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All singletons {p} except {O} are algebraic characteristic sets, specifically for the Coordinatizations 25 Lazerson matroids, which, as we have noted, also have {P} for their linear characteristic sets (Lindstrom 1985c). A number of finite, non-singleton, algebraic characteristic sets are known (Gordon 1987). It is not hard to show the following proposition (see Welsh 1976, p. 187). 7. Proposition. If M(S) is algebraic over F and AsS, then the contraction MIA is algebraic over a transcendental extension of F, and hence over F.

Montreal, Montreal 1979) Part I Ann. Discrete Math. 8. (1980), 83-90. T. (1965). Lectures on matroids. J. Res. Nat. Bur. Stand. 698, \-47. A. (1969a). On the hyperplanes of a matroid. Proc. Cambridge Phil. Soc. 65, l1-18. A. (1969b). Euler and bipartite matroids. J. Comb. Theory, 6,375-7. White, N. (1971). The Bracket ring and combinatorial geometry. , ed. (1986). Theory of Matroids, Cambridge University Press. Whitney H. (1933). Planars graphs. Fund. Math 21, 73-84. Whitney H. (1935). On the abstract properties of linear dependence.

7 Appendix on Modular Pairs of Circuits in a Matroid Two circuits C l and C 2 in a matroid M(E), in which ris the rank function, form a modular pair if they satisfy the modular relation: This relation implies, when C 1 "1= C 2 , that We then verify easily that given two distinct circuits C l and C2 in M(E), and looking at the hyperplanes Hi = E\Cl,Hi = E\C 2 in M*(E), we have: C 1 and C 2 form a modular pair if and only if Hi n Hi is a coline in M*(E). 1. Lemma. (White 1971) Let C 1 and C 2 be a modular pair ofcircuits such that C 1 "1= C 2 , C 1 n C 2 "1= 0, and let a be an element of C 1 n ;:: l.