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Lattices with Doubly Irreducible Elements

Published online by Cambridge University Press:  20 November 2018

Ivan Rival
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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
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3. Havas, G. and Ward, M., Lattices with sublattices of a given order, J. Combinatorial Theory 7 (1969), 281-282.Google Scholar