Free Lattices

Freese, Ralph; Ježek, Jaroslav; Nation, J. B.

doi:10.1090/surv/042

Cited by 195 publications

(311 citation statements)

References 11 publications

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“…For example the lattices associated with (x ∨ (y ∧ z)) ∧ (y ∨ z) and with (x ∨ (y ∧ z)) ∧ (y ∨ (x ∧ z)) are diagrammed in Figure 5. (Since the second lattice is semidistributive but the first is not, (x ∨ (y ∧ z)) ∧ (y ∨ z) has a lower cover but (x ∨ (y ∧ z)) ∧ (y ∨ (x ∧ z)) does not; see [10]). …”

Section: Automatic Drawingmentioning

confidence: 99%

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Automated Lattice Drawing

Freese

2004

Concept Lattices

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Abstract. Lattice diagrams, known as Hasse diagrams, have played an ever increasing role in lattice theory and fields that use lattices as a tool. Initially regarded with suspicion, they now play an important role in both pure lattice theory and in data representation. Now that lattices can be created by software, it is important to have software that can automatically draw them. This paper covers:-The role and history of the diagram.-What constitutes a good diagram.-Algorithms to produce good diagrams. Recent work on software incorporating these algorithms into a drawing program will also be covered.An ordered set P = (P, ≤) consists of a set P and a partial order relation ≤ on P . That is, the relation ≤ is reflexive (x ≤ x), transitive (x ≤ y and y ≤ z imply x ≤ z) and antisymmetric (x ≤ y and y ≤ x imply x = y). If P is finite there is a unique smallest relation ≺, known as the cover or neighbor relation, whose transitive, reflexive closure is ≤. (Graph theorists call this the transitive reduct of ≤.) A Hasse diagram of P is a diagram of the acyclic graph (P, ≺) where the edges are straight line segments and, if a < b in P, then the vertical coordinate for a is less than the one for b. Because of this second condition arrows are omitted from the edges in the diagram.A lattice is an ordered set in which every pair of elements a and b has a least upper bound, a ∨ b, and a greatest lower bound, a ∧ b, and so also has a Hasse diagram.These Hasse diagrams 1 are an important tool for researchers in lattice theory and ordered set theory and are now used to visualize data. This paper deals the special issues involved in such diagrams. It gives several approaches that have been used to automatically draw such diagrams concentrating on a three dimension force algorithm especially adapted for ordered sets that does particularly well.We begin with some examples.1 In the second edition of his famous book on lattice theory [3] Birkhoff says these diagrams are called Hasse diagrams because of Hasse's effective use of them but that they go back at least to H. Vogt, Résolution algébrique deséquation, Paris, 1895.

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Section: Automatic Drawingmentioning

confidence: 99%

“…The usual algorithm to find a linear extension of P (graph theorists call this topological sorting) can be easily modified to calculate this rank function in linear time. (See Chapter 11 of [10] for a discussion of algorithms for ordered sets and lattices. )…”

Section: The Rank Functionmentioning

confidence: 99%

Automated Lattice Drawing

Freese

2004

Concept Lattices

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“…There may not be a b so that (a, b) is critical. Although it is phrased differently, [12,Theorem 2.56] states that the meet semi-distributive property of a lattice is equivalent to the property that for any join-irreducible a there is a unique b so that (a, b) is subcritical. The dual result also holds.…”

Section: By Theorem 6 S(y ∨ Z) = S(y ) ∪ S(z)mentioning

confidence: 99%

Lattice and order properties of the poset of regions in a hyperplane arrangement

Reading

2003

Algebra Universalis

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Abstract. We show that the poset of regions (with respect to a canonical base region) of a supersolvable hyperplane arrangement is a congruence normal lattice. Specifically, the poset of regions of a supersolvable arrangement of rank k is obtained via a sequence of doublings from the poset of regions of a supersolvable arrangement of rank k − 1. An explicit description of the doublings leads to a proof that the order dimension of the poset of regions (again with respect to a canonical base region) of a supersolvable hyperplane arrangement is equal to the rank of the arrangement. In particular, the order dimension of the weak order on a finite Coxeter group of type A or B is equal to the number of generators. The result for type A (the permutation lattice) was proven previously by Flath [11].We show that the poset of regions of a simplicial arrangement is a semi-distributive lattice, using the previously known result [2] that it is a lattice. A lattice is congruence uniform (or "bounded" in the sense of McKenzie [18]) if and only if it is semi-distributive and congruence normal [7]. Caspard, Le Conte de Poly-Barbut and Morvan [4] showed that the weak order on a finite Coxeter group is congruence uniform. Inspired by the methods of [4], we characterize congruence normality of a lattice in terms of edge-labelings. This leads to a simple criterion to determine whether or not a given simplicial arrangement has a congruence uniform lattice of regions. In the case when the criterion is satisfied, we explicitly characterize the congruence lattice of the lattice of regions. Main ResultsWe begin by listing the main results, with most definitions put off until Section 2. Theorem 1. The poset of regions (with respect to a canonical base region) of a supersolvable hyperplane arrangement is a congruence normal lattice.A lattice is called congruence normal if it can be obtained from the one-element lattice by a sequence of doublings of convex sets [6]. The doublings in Theorem 1 correspond to adjoining hyperplanes one by one to form the arrangement. Björner, Edelman and Ziegler [2] showed that the poset of regions (with respect to a canonical base region) of a supersolvable hyperplane arrangement is a lattice. The proof given here of Theorem 1 provides a different proof of the lattice property, but uses several key observations made in [2].

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“…We recall the representation of congruence relations on a finite lattice in terms of the dependency relation; see, e.g., [4], pages 40-41.…”

Section: There Is a Lattice Embeddingmentioning

confidence: 99%