Embeddings of Finite Distributive Lattices into Products of Chains

Larson, Rebecca N.

doi:10.1007/s00233-002-7005-3

Cited by 4 publications

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“…Indeed, intervals I(p, q) of median graphs and median semilattices can be viewed as distributive lattices by setting x ∧ y = m(p, x, y) and x ∨ y = m(q, x, y) for any x, y ∈ I(p, q), where m is the median operator of G [6,10]. Using the encoding of distributive lattices via closed subsets of a poset due to Birkhoff [9], the famous Dilworth's theorem (the size of a largest antichain of a poset equals to the least size of a decomposition of the poset into chains) [21] implies that any distributive lattice L of breadth k can be embedded as a sublattice of a product of k chains, see [35] or [19] for this interpretation of Dilworth's result (the breadth of a distributive lattice L is equal to the largest out-or in-degree of a vertex in the covering graph of L). Larson [35] showed that the resulting embedding can be chosen to preserve the covering relation, i.e.…”

Section: Proofmentioning

confidence: 99%

Shortest path problem in rectangular complexes of global nonpositive curvature

Chepoi

Maftuleac

2013

Computational Geometry

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CAT(0) metric spaces constitute a far-reaching common generalization of Euclidean and hyperbolic spaces and simple polygons: any two points x and y of a CAT(0) metric space are connected by a unique shortest path γ(x, y). In this paper, we present an efficient algorithm for answering two-point distance queries in CAT(0) rectangular complexes and two of theirs subclasses, ramified rectilinear polygons (CAT(0) rectangular complexes in which the links of all vertices are bipartite graphs) and squaregraphs (CAT(0) rectangular complexes arising from plane quadrangulations in which all inner vertices have degrees ≥ 4). Namely, we show that for a CAT(0) rectangular complex K with n vertices, one can construct a data structure D of size O(n 2 ) so that, given any two points x, y ∈ K, the shortest path γ(x, y) between x and y can be computed in O (d(p, q)) time, where p and q are vertices of two faces of K containing the points x and y, respectively, such that γ(x, y) ⊂ K(I(p, q)) and d(p, q) is the distance between p and q in the underlying graph of K. If K is a ramified rectilinear polygon, then one can construct a data structure D of optimal size O(n) and answer two-point shortest path queries in O(d(p, q) log ∆) time, where ∆ is the maximal degree of a vertex of G(K).Finally, if K is a squaregraph, then one can construct a data structure D of size O(n log n) and answer two-point shortest path queries in O(d(p, q)) time.

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Section: Proofmentioning

confidence: 99%

Shortest path problem in rectangular complexes of global nonpositive curvature

Chepoi

Maftuleac

2013

Computational Geometry

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“…The image of L by the embedding is marked out. Alternatively, L can always be seen as a sublattice of a product of chains [22]. Let k be a positive integer and C 1 , .…”

Section: Basic Background On Distributive Latticesmentioning

confidence: 99%

Interpolation by lattice polynomial functions: A polynomial time algorithm

Brabant

Couceiro

Figueira

2019

Fuzzy Sets and Systems

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This paper deals with the problem of interpolating partial functions over finite distributive lattices by lattice polynomial functions. More precisely, this problem can be formulated as follows: Given a finite distributive lattice L and a partial function f from D ⊆ L n to L, find all the lattice polynomial functions that interpolate f on D. If the set of lattice polynomials interpolating a function f is not empty, then it has a unique upper bound and a unique lower bound. This paper presents a new description of these bounds and proposes an algorithm for computing them that runs in polynomial time, thus improving existing methods. Furthermore, we present an empirical study on randomly generated datasets that illustrates our theoretical results.

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“…Theorem 2.5. [6] The embeddings e C of Corollary 2.4 are tight, and for every tight embedding e there is a chain decomposition C of J(L) such that e = e C .…”

Section: Notation and Backgroundmentioning

confidence: 99%

On the representation of finite distributive lattices

Siggers

2014

Preprint

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A simple but elegant result of Rival states that every sublattice L of a finite distributive lattice P can be constructed from P by removing a particular family I L of its irreducible intervals.Applying this in the case that P is a product of a finite set C of chains, we get a one-to-one correspondence L → D P (L) between the sublattices of P and the preorders spanned by a canonical sublattice C ∞ of P.We then show that L is a tight sublattice of the product of chains P if and only if D P (L) is asymmetric. This yields a one-to-one correspondence between the tight sublattices of P and the posets spanned by its poset J(P) of non-zero join-irreducible elements.With this we recover and extend, among other classical results, the correspondence derived from results of Birkhoff and Dilworth, between the tight embeddings of a finite distributive lattice L into products of chains, and the chain decompositions of its poset J(L) of non-zero join-irreducible elements.2010 Mathematics Subject Classification. 06D05.

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Embeddings of Finite Distributive Lattices into Products of Chains

Cited by 4 publications

References 0 publications

Shortest path problem in rectangular complexes of global nonpositive curvature

Shortest path problem in rectangular complexes of global nonpositive curvature

Interpolation by lattice polynomial functions: A polynomial time algorithm

On the representation of finite distributive lattices

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