Let $M$ be a matroid on a finite set $E$ and $A\subset E$. I want to define a span of $A$, but not as a subset of $M$ (which could be defined as a union of all $B\supset A$ satisfying ${\rm rank}\, B={\rm rank}\, A$): I would like to consider elements not lying in $M$. This can be defined as a lattice of classes of extensions of $M$ to matroids on larger finite sets $E\sqcup E_1$ satisfying ${\rm rank}\, A\sqcup E_1={\rm rank}\, A$. If $M$ is representable over a field (and all extensions which we consider also are), this is like a genuine span, at least in the sense of matroid structures on finite subsets. Are there references to such a notion? Is there another, more direct definition?
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asked Apr 19, 2017 at 5:59
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$\begingroup$ Would you please be more specific what data your span-operator should take as its input? The standard span operator in matroid theory, which of course can be defined in different equivalent ways, takes as input (0) a matroid w.r.t. to which the span is taken, and (1) a subset of which the span is taken. And as to your question about more "direct" (interpreting this as: not using a rank-function) definitions, this hint may help: it is very common to define spans combinatorially as 'the set of all elements outside such that when adding the element there is a circuit contained in the result". $\endgroup$– Peter HeinigCommented Apr 21, 2017 at 15:49
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$\begingroup$ I want to define span which is not as a subset of matroid. An analogue of linear span for vector matroids. $\endgroup$– Fedor PetrovCommented Apr 21, 2017 at 16:01
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1$\begingroup$ I've seen something along these lines for oriented matroids in the following book by Santos: personales.unican.es/santosf/Articulos/OMtriFinal.pdf see Definition 2.1, part "intersect properly". He treats the set of all extensions satisfying certain properties as a genuine convex hull of a simplex. $\endgroup$– Pavel GalashinCommented Apr 25, 2017 at 10:28
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