Isometrical Embeddings of Lattices into Geometric Lattices

Skublics, Benedek

doi:10.1007/s11083-012-9277-x

Cited by 5 publications

(9 citation statements)

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“…In the first section, we recall a proof of Marcel Wild [89], which shows that every finite semimodular lattice has a cover-preserving embedding into a geometric lattice. This argument is a motivation for the second section, where we prove a generalization of an embedding result of George Grätzer and Emil W. Kiss [43], see [78].…”

Section: Discussionmentioning

confidence: 89%

“…However, if a lattice is of finite length and both itself and its dual are semimodular then it is also modular. The three chapters of my dissertation are based on the papers [27,78] and [77].…”

Section: Discussionmentioning

confidence: 99%

“…In [78] we managed to show that Grätzer and Kiss' theorem can also be extended for lattices of finite length, moreover, it can be extended for a larger class of lattices that we called finite height generated lattices.…”

Section: Chapter 2 Isometrical Embeddingsmentioning

confidence: 99%

“…Wild [89] matroidokkal történő fedésőrző beágyazása található véges féligmoduláris hálókra, ami motivációt ad azáltalános eset bizonyításához, amit a fejezet második részében közlök [78].…”

Section: Mal'cev Conditionsunclassified

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Modular and semimodular lattices

Skublics

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Section: Discussionmentioning

confidence: 89%

Section: Discussionmentioning

confidence: 99%

Section: Chapter 2 Isometrical Embeddingsmentioning

confidence: 99%

Section: Mal'cev Conditionsunclassified

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Modular and semimodular lattices

Skublics

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“…One question revolved around whether or not such lattices need to have atomic elements. It turns out that they do not, and we use Ann-Kathrin Elm's formulation the construction found in [27] to clearly identify a countable matroid which does not possess an atomic element. We therefore say that a matroid is atomless if its lattice of cyclic flats has no atomic elements, and call the matroid weakly atomic if there is an atomic element in its lattice of cyclic flats.…”

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confidence: 99%

Infinite Matroids and Transfinite Sequences

Storm¹

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Infinite Matroids and Transfinite Sequences Martin Storm A matroid is a pair M = (E, I) where E is a set and I is a set of subsets of E that are called independent, echoing the notion of linear independence. One of the leading open problems in infinite matroid theory is the Matroid Intersection Conjecture by Nash-Williams which is a generalization of Hall's Theorem. In [31] Jerzy Wojciechowski introduced µ-admissibility for pairs of matroids on the same ground set and showed that it is a necessary condition for the existence of a matching. A pair of matroids (M, W) with common ground set E is µ-admissible if a subset of sequences in E × {0, 1} have a certain property. In order to determine if this property implies anything about the length of the sequence, we modify µ-admissibility to obtain µ-admissibility, an equivalent property for pairs of matroids. We then use µ-admissibility to show that for every successor ordinal of the form α + 2n there is a pair of partition matroids such that a shortest sequence in E × {0, 1} that fails to have the desired property has length α + 2n. Furthermore, we introduce the class of patchwork matroids, which contains all finite matroids and all uniform matroids, and provide a method for their construction, prove a characterization theorem, show the class is closed under duality and taking minors, as well as several other properties. Lastly, a cyclic flat is a flat which is a union of circuits. We combine this notion with that of trees of matroids, which are introduced in [7] to show that the lattice of cyclic flats of a locally finite tree of finite matroids contains an atomic element.

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