By Neil White

A continuation of the speculation of Matroids, (edited by way of N. White), this quantity contains a chain of similar surveys by means of most sensible professionals on coordinatizations, matching idea, transversal and simplicial matroids, and stories of vital matroid editions. a complete bankruptcy is dedicated to matroids in combinatorial optimization, a subject of present curiosity. Care has been taken to make sure a uniform variety all through, and to make a piece that may be used as a reference or as a graduate textbook. Excercises are incorporated.

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Thus if M(E) is connected, we have the following characterization: (4') Let e be an element in the set E. Given any two distinct circuits C l , C 2 not containing e, then C l 6. C 2 contains a circuit. Condition (6) can be weakened as follows (Seymour 1976): (6') For any circuit C and any cocircuit C*, ICnC*1 #3. The last following characterization is stated in terms of a new property of circuits (Fournier 1974b). We say that a circuit C l distinguishes circuits C2 and C3 if C 2\C l # C 3 \C l .

Gordon, G. (1987). Algebraic characteristic sets ofmatroids. J. Comb. Theory Ser. B. to appear. W. (1971). 149-69. Academic Press, London and New York. Ingleton, A. and Main, R. (1975). Non-algebraic matroids exist. London Math. Soc. 7, 144-6. Kahn, J. (1981). Characteristic sets ofmatroids. Dennison-OSU Math. Conference, Dennison, Ohio. M. (1975). Envelopes of geometric lattices. J. Comb. Theory Ser. A 18, 12-26. Lang, S. (1965). Algebra, Addison-Wesley. Lazarson, T. (1958). The representation problem for independence functions.

This completes the proof of the theorem. D The proofs of equivalence we have given are all direct and make no appeal to other theorems concerning the representation of matroids. 5) would, for example, have given directly the implication (5)=>(0). 1) would give directly the equivalence of conditions (7) and (8), these conditions being interpreted in the language of lattice theory, as indicated above for condition (7). Finally, we note that one can see directly that the matroid U 2,4' that unique smallest non-binary matroid, in the sense of condition (7), is not binary, because the projective line over the field GF(2) has only three points, whereas U 2,4 has four points.