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The LCMlattice in monomial resolutions
, 1999
"... Describing the properties of the minimal free resolution of a monomial ideal I is a difficult problem posed in the early 1960’s. The main directions of progress on this problem were: ..."
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Cited by 38 (5 self)
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Describing the properties of the minimal free resolution of a monomial ideal I is a difficult problem posed in the early 1960’s. The main directions of progress on this problem were:
Convex drawings of Planar Graphs and the Order Dimension of 3Polytopes
 ORDER
, 2000
"... We define an analogue of Schnyder's tree decompositions for 3connected planar graphs. Based on this structure we obtain: Let G be a 3connected planar graph with f faces, then G has a convex drawing with its vertices embedded on the (f 1) (f 1) grid. Let G be a 3connected planar graph. ..."
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Cited by 32 (13 self)
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We define an analogue of Schnyder's tree decompositions for 3connected planar graphs. Based on this structure we obtain: Let G be a 3connected planar graph with f faces, then G has a convex drawing with its vertices embedded on the (f 1) (f 1) grid. Let G be a 3connected planar graph. The dimension of the incidence order of vertices, edges and bounded faces of G is at most 3. The second result is originally due to Brightwell and Trotter. Here we give a substantially simpler proof.
The order dimension of planar maps
 SIAM J. DISCRETE MATH
, 1997
"... This is a sequel to a previous paper entitled The Order Dimension of Convex Polytopes, by the same authors [SIAM J. Discrete Math., 6 (1993), pp. 230–245]. In that paper, we considered the poset PM formed by taking the vertices, edges, and faces of a 3connected planar map M, ordered by inclusion, ..."
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Cited by 12 (4 self)
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This is a sequel to a previous paper entitled The Order Dimension of Convex Polytopes, by the same authors [SIAM J. Discrete Math., 6 (1993), pp. 230–245]. In that paper, we considered the poset PM formed by taking the vertices, edges, and faces of a 3connected planar map M, ordered by inclusion, and showed that the order dimension of PM is always equal to 4. In this paper, we show that if M is any planar map, then the order dimension of PM is still at most 4.
A Homological Lower Bound For Order Dimension Of Lattices
 ORDER
"... We prove that if a finite lattice L has order dimension at most d, then the homology of the order complex of its proper part L ffi vanishes in dimensions d  1 and higher. In case L can be embedded as a joinsublattice in N d then L ffi actually has the homotopy type of a simplicial complex with d v ..."
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Cited by 1 (0 self)
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We prove that if a finite lattice L has order dimension at most d, then the homology of the order complex of its proper part L ffi vanishes in dimensions d  1 and higher. In case L can be embedded as a joinsublattice in N d then L ffi actually has the homotopy type of a simplicial complex with d vertices.
On multipartite posets
, 2007
"... A poset P = (X, ≼) is mpartite if X has a partition X = X1 ∪ · · · ∪ Xm such that (1) each Xi forms an antichain in P, and (2) x ≺ y implies x ∈ Xi and y ∈ Xj where i < j. In this article we derive a tight asymptotic upper bound on the order dimension of mpartite posets in terms of m and th ..."
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A poset P = (X, ≼) is mpartite if X has a partition X = X1 ∪ · · · ∪ Xm such that (1) each Xi forms an antichain in P, and (2) x ≺ y implies x ∈ Xi and y ∈ Xj where i < j. In this article we derive a tight asymptotic upper bound on the order dimension of mpartite posets in terms of m and their bipartite subposets in a constructive and elementary way.