Hostname: page-component-77c89778f8-gq7q9 Total loading time: 0 Render date: 2024-07-16T14:36:24.738Z Has data issue: false hasContentIssue false
  • English
  • Français

Lattices with Doubly Irreducible Elements

Published online by Cambridge University Press:  20 November 2018

Ivan Rival
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

An element x in a lattice L is join-reducible (meet-reducible) in L if there exist y, z∈L both distinct from x such that x=y⋁z (x=y⋀z); x is join-irreducible (meet-irreducible) in L if it is not join-reducible (meet-reducible) in L; x is doubly irreducible in L if it is both join- and meet-irreducible in L. Let J(L), M(L), and Irr(L) denote the set of all join-irreducible elements in L, meet-irreducible elements in L, and doubly irreducible elements in L, respectively, and ℓ(L) the length of L, that is, the order of a maximum-sized chain in L minus one.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 17 , Issue 1 , 01 March 1974 , pp. 91 - 95
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Baker, K. A., Fishburn, P. C., and Roberts, F. S., Partial orders of dimension 2, interval orders, and interval graphs, The Rand Corp., P-4376 (1970).Google Scholar
2. Grätzer, G., Lattice theory: First concepts and distributive lattices, Freeman, W. H., San Francisco, 1971.Google Scholar
3. Havas, G. and Ward, M., Lattices with sublattices of a given order, J. Combinatorial Theory 7 (1969), 281-282.Google Scholar