On geometric semilattices
type: Item
instance of: scholarly article
title: On geometric semilattices
zbMATH Open document ID: 0589.06005
DOI: 10.1007/BF00367425
author: Michelle L. Wachs
author: James W. Walker
published in: Order
publication date: 1986-01-01T00:00:00Z
review text: A geometric semilattice is a ranked meet semilattice such that every element is the join of atoms and such that the collection of independent sets of atoms forms a matroid. The class of ranked lattices with these properties is the class of geometric lattices. Other examples of interest include the set of upper semicomplements of a fixed element in the lattice of linear subspaces of a finite-dimensional vector space as well as the set of affine subspaces of a finite-dimensional vector space which do not contain the origin. The authors established enough general theory in this paper to allow one to use it as a starting point for the entire complex of notions related to shellability, Cohen-Macaulayness and special properties of the Möbius function with which the authors as well as the editor (A. Björner) are associated among others. In passing they prove a good sized collection of interesting lemmas and propositions which demonstrate the strength and usefulness of the notion introduced. For considerations which are more categorical in nature they extend the notion of strong map from geometric lattices to geometric semilattices, which allows them to be compared to their lattices of flats. An important construction is that of a mapping cylinder. Using these geometric lattices have nice characterizations, e.g., as geometric lattices with intervals [x,1] removed. Using these characterizations shellability and the Möbius functions are well within reach also.
Mathematics Subject Classification ID: 06C10
Mathematics Subject Classification ID: 06A12
Mathematics Subject Classification ID: 05B35
zbMATH DE Number: 3946192
zbMATH Keywords: ranked meet semilattice
zbMATH Keywords: join of atoms
zbMATH Keywords: matroid
zbMATH Keywords: ranked lattices
zbMATH Keywords: geometric lattices
zbMATH Keywords: upper semicomplements
zbMATH Keywords: shellability
zbMATH Keywords: Cohen- Macaulayness
zbMATH Keywords: Möbius function
zbMATH Keywords: strong map
reviewed by: J. Neggers
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00367425
OpenAlex ID: W2001432829
description: scientific article; zbMATH DE number 3946192