Poset Associahedra
Patrick Showers
October 18, 2012
3 a 3 0 c c a 2 c a a f 1AB 2 1 3 0 b 1 2 b a b 2 b b 2 A B d d 1 1 4
a b c a c d f a b b a b c d a b
0 1 2 3 0 1 2 3 1 2 1 2 1 2 3 4
Patrick Showers Poset Associahedra But first...
Preview
Polytope KP is a lattice of tubings, which are in turn comprised of tubes.
Patrick Showers Poset Associahedra Preview
Polytope KP is a lattice of tubings, which are in turn comprised of tubes.
But first...
Patrick Showers Poset Associahedra Predefinitions
I A lower set S of a poset contains all elements {x | x ≤ s, s ∈ S}.
Patrick Showers Poset Associahedra Lower Sets
Lower sets contain everything ‘beneath’ their members
Patrick Showers Poset Associahedra Predefinitions
I A lower set S of a poset contains all elements {x | x ≤ s, s ∈ S}.
I Important example of lower set:
I The boundary ∂p of a poset element p consisting of all elements strictly less than p.
Patrick Showers Poset Associahedra Boundaries
The boundary of an element is everything strictly below that element
Patrick Showers Poset Associahedra Utilizing Boundaries
I If we fix a poset element p and ask what other elements q satisfy ∂q = ∂p, we have the bundle bp = {q | ∂q = ∂p}
Patrick Showers Poset Associahedra Bundles
A B C ∂A = ∂B = ∂C
a b ∂a = ∂b
1 2 ∂1 = ∅ = ∂2
Patrick Showers Poset Associahedra Utilizing Boundaries
I If we fix a poset element p and ask what other elements q satisfy ∂q = ∂p, we have the bundle bp = {q | ∂q = ∂p}
I With boundaries, we call a subset S of poset elements filled if bp ∩ S 6= ∅ whenever ∂p ⊆ S
Patrick Showers Poset Associahedra Filled Sets
Filled sets have at least one p if they include ∂p
Patrick Showers Poset Associahedra Filled Sets
Filled sets have at least one p if they include ∂p
Patrick Showers Poset Associahedra Tubes and Tubings
A tube of a poset is a filled, connected, lower set
a b c d
Figure 1. Here is a Hasse diagram of a poset with various lower sets circled.(a) and (b) are not valid tubings, since both fail to be lled. (c) and (d) are tubings.
Patrick Showers Poset Associahedra and
I the union of any subset of tubes is filled
Tubes and Tubings
I A tube of a poset is a filled, connected, lower set
) a ( a ) ( b ) ( c ) ( d )
I A tubing is a set of tubes such that
I all tubes are pairwise nested or pairwise disjoint,
Patrick Showers Poset Associahedra Tubes and Tubings
I A tube of a poset is a filled, connected, lower set
) a ( a ) ( b ) ( c ) ( d )
I A tubing is a set of tubes such that
I all tubes are pairwise nested or pairwise disjoint, and
I the union of any subset of tubes is filled
Patrick Showers Poset Associahedra The Poset Associahedron
Given poset P:
The poset associahedron KP is the lattice of tubings, ordered by reverse inclusion
Patrick Showers Poset Associahedra Examples of KP
Associahedra
Patrick Showers Poset Associahedra Examples of KP
Permutohedra
Patrick Showers Poset Associahedra Abstract Polytope
An abstract polytope is a poset P satisfying four axioms (i) P has a least and greatest face (ii) Flags of P are the same length (iii) P is strongly-connected (iv) 1-sections of P are isomorphic to line segments
A strongly-connected lattice whose 1-sections are line-segments
Patrick Showers Poset Associahedra KP as Abstract Polytope
I Theorem KP is an abstract polytope. Further,
I KP is simple P I The dimension of KP is (|ba| − 1)
Patrick Showers Poset Associahedra Conjecture
KP is a convex polytope
12 bx 12 bcx
12 bc a b c x
2x
1 2 12 ax
2c 2cx
12 abc 12 ac
Patrick Showers Poset Associahedra More Examples!
f = 32, 64, 38, 12
Patrick Showers Poset Associahedra More Examples!
f = 68, 136, 88, 20 3d facet with 2 octagons
Patrick Showers Poset Associahedra a
Patrick Showers Poset Associahedra