Jamison Abstract
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Monotaxis in Matroids and Antimatroids
Robert Jamison
Department of Mathematical Sciences
Clemson University
Clemson, SC 29634-1907
rejam@clemson.edu
OFFICE: (803) 656-5219
FAX: (803) 656-5230
Matroids are closure systems which satisfy the exchange law and
abstractly model the combinatorial properties of linear span.
Anti-matroids are closure systems which satisfy the anti-exchange law
and abstractly model the
combinatorial properties of convex hull. Very roughly speaking, a
closure
system is monotactic iff it possess a unique "simple" procedure for
testing
the closure of a set. For example, for a subset of Euclidean space to
be
linearly closed, it suffices that it contain the linear span of any
two of its
points. Likewise, for a subset of Euclidean space to be convex, it
suffices
that it contain the line segment between any two of its points.
There are no other equally "simple" descriptions of these closures and
hence these closures are both monotactic. In general, most "complete"
natural closures tend to be
monotactic. The notion becomes of interest, however, when one
considers
hereditarily monotactic spaces -- those in which ALL subspaces are
monotactic.
Recent work by Jamison & Oxley show that the class of matroids all
of whose minors are monotactic is only slightly larger than the class
of binary
matroids. Earlier work by Jamison & Strahringer established that the
class of
monotactic anti-matroids is much richer -- including trees,
semilattices, and
their disjunctive products.
The talk will, of course, provide careful definitions for the
technical terms used above.
Thu October 12 11:50:14 EDT 1995