Modular and distributive semilattices
Author:
Joe B. Rhodes
Journal:
Trans. Amer. Math. Soc. 201 (1975), 31-41
MSC:
Primary 06A20
DOI:
https://doi.org/10.1090/S0002-9947-1975-0351935-X
MathSciNet review:
0351935
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Abstract | References | Similar Articles | Additional Information
Abstract: A modular semilattice is a semilattice in which
implies that there exist
such that
and
. This is equivalent to modularity in a lattice and in the semilattice of ideals of the semilattice, and the condition implies the Kurosh-Ore replacement property for irreducible elements in a semilattice. The main results provide extensions of the classical characterizations of modular and distributive lattices by their sublattices: A semilattice
is modular if and only if each pair of elements of
has an upper bound in
and there is no retract of
isomorphic to the nonmodular five lattice. A semilattice is distributive if and only if it is modular and has no retract isomorphic to the nondistributive five lattice.
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1975-0351935-X
Keywords:
Lattice,
semilattice,
poset,
modular,
distributive,
semimodular,
ideal,
retract,
neet-irreducible,
finite maximal chain
Article copyright:
© Copyright 1975
American Mathematical Society