Introduction
Matroids (see [27]) and oriented matroids (see [11]) are axiomatic abstract
models for combinatorial geometry over general fields and over ordered fields, res-
pectively. Oriented matroids have some extra structure as compared to matroids,
one of whose features is the existence of a notion of convexity (see Chapter 9 of
[11]). This makes it natural to consider triangulations of oriented matroids as an
analogue of triangulations of usual polytopes or point configurations. This concept
is the object of this paper.
Triangulations of oriented matroids generalize the following situations, where
the oriented matroids involved are realized by geometric objects and where the
triangulations of the geometric objects considered are known to depend only on the
underlying oriented matroid:
If M. is the oriented matroid of affine dependences between the vertices
of a polytope P, the triangulations of M coincide with the triangulations
of the polytope P, meaning by this the geometric simplicial complexes
which cover P and use only the vertices of P as vertices. There is a recent
survey by Lee [25] on this topic.
If M. is the oriented matroid of affine dependences between the points
in a finite point set A in Rd, the triangulations of M coincide with the
triangulations of A, meaning by this the geometric simplicial complexes
which cover the convex hull of A and which use (perhaps not all) the
points of A as vertices. This is a generalization of the previous case
which has been often considered in recent literature (see [6, 13, 14, 15,
16, 28, 30, 36] and Chapter 7 of [19]).
If M. is the oriented matroid of linear dependences between a finite set of
vectors V in
Rd,
the triangulations of M coincide with the triangulations
of V, meaning by this the simplicial fans covering the positive span of V
and whose rank-1 cones are generated by (perhaps not all) the vectors of
A- See [19, Definition 4.1] or [7]. This is a further generalization of the
previous case: if A is a point configuration in Rd, then triangulations of
A coincide with the simplicial fans of the vector configuration obtained
by embedding Rd as an affine hyperplane in the vector space Rrf+1.
Triangulations of oriented matroids were first defined by Billera and Munson
in [9], for the special case of polytopal oriented matroids. An account of them for
the more general case of acyclic oriented matroids appears in Section 9.6 of [11].
Deciding which is the best, or natural, definition of triangulation of a general
oriented matroid is not trivial. Apart of the problem of translating geometric
conditions into oriented matroid language, different possible characterizations of
triangulations in the realized case may translate to non-equivalent definitions in
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