Does this definition of dim 2 n-free poset "exterior" ring any bells with anyone?
From: David Halitsky (dhalitsky_at_cumulativeinquiry.com)
Date: 10/22/04
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Date: 22 Oct 2004 10:01:49 -0400
Let H be the Hasse diagram of any dim 2 n-free poset
and let Hp be any natural plane embedding of H. Then
inasmuch as Hp is a DAG whose edges are defined by the
immediate covering relations in H, one can assign
the integer d to each vertex v of Hp, where d is the
depth of v in Hp interpreted as a DAG (i.e. the number
of vertices in Hp which are ancestors of v, excluding v
itself.)
Further, since Hp is embedded in the plane "relative
to the geometry of the paper", a "left-right" or
"horizontal" ordering of the vertices of Hp results
from its embedding in the plane. That is, we can
assign the numbers p and t to each vertex v of Hp, where:
p is the number of vertices "to the left" of v in Hp
t is the number of vertices "to the right" of v in Hp.
And it is readily shown that because Hp is both dim 2
and n-free:
a) the values (p0+d0),...,(pi+di),...,(pn+dn) for
the n+1 vertices of Hp are 0,...,n+1
b) the values (t0+d0),...,(ti+di),...,(tn+dn) for
the n+1 vertices of Hp are a permutation of 0,...,n+1.
Suppose now, however, that one uses the integers
pi, ti, di to assign each vertex v of Hp the triple
of integers (pi,ti,di).
And further, suppose that one defines the exterior
of Hp as just those vertices in Hp for which at least
one of pi,ti,di is 0.
And finally, suppose that one defines the exterior
union of Hp as the union of all the exteriors of all
the subgraphs of Hp which are themselves interpretable
as Hasse diagrams of dim 2 n-free posets.
Does this notion of "exterior union" ring any bells
with anyone ??
Thanks for whatever time anyone can afford to spend
considering this matter.
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