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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Modular and distributive semilattices

Author: Joe B. Rhodes
Journal: Trans. Amer. Math. Soc. 201 (1975), 31-41
MSC: Primary 06A20
MathSciNet review: 0351935
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Abstract: A modular semilattice is a semilattice $ S$ in which $ w \geq $ implies that there exist $ x,y \in S$ such that $ x \geq a,y \geq b$ and $ x \wedge y = x \wedge w$. 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 $ S$ is modular if and only if each pair of elements of $ S$ has an upper bound in $ S$ and there is no retract of $ S$ 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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Keywords: Lattice, semilattice, poset, modular, distributive, semimodular, ideal, retract, neet-irreducible, finite maximal chain
Article copyright: © Copyright 1975 American Mathematical Society