Modular and distributive semilattices
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- by Joe B. Rhodes PDF
- Trans. Amer. Math. Soc. 201 (1975), 31-41 Request permission
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.References
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Additional Information
- © Copyright 1975 American Mathematical Society
- 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