Abstract
A natural sufficient condition for a finite family of single element extensions of a matroid to be compatible is given. Characterizations of all the finite extensions N of a matroid M( E) are given for which the rank function satisfies ρ N(X)= Min Z⊂E {ρ M(Z)+|X− Z N|} or equivalently the closure operator satisfies X N = X N ⌢ E N ⌣ X . The single element extensions and the principal extensions are examples of such matroids. The notion of a sheaf of flats of M. Las Vergnas is used in the proof of a new necessary and sufficient condition for two single element extensions of a matroid to be compatible. An initial announcement of part of these results appeared in R. Cordovil ( C. R. Acad. Sci. Paris. A 284 (1977), 1249–1252).
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