Join-principal element lattices

Johnson, E. W.

doi:10.1090/S0002-9939-1976-0404067-2

Join-principal element lattices
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by E. W. Johnson PDF

Proc. Amer. Math. Soc. 57 (1976), 202-204 Request permission

Abstract:

Let $(\mathfrak {L},M)$ be a local Noether lattice. If the maximal element $M$ is meet principal, it is well known and easily seen that every element of $\mathfrak {L}$ is meet principal. In this note, we obtain the corresponding result for $M$ join-principal. We also consider join-principal elements generally under the assumption of the weak union condition and show, for example, that the square of a join-principal element is principal.

References

E. W. Johnson and J. P. Lediaev, Structure of Noether lattices with join-principal maximal elements, Pacific J. Math. 37 (1971), 101–108. MR 307993
E. W. Johnson and J. P. Lediaev, Join-principal elements in Noether lattices, Proc. Amer. Math. Soc. 36 (1972), 73–78. MR 306174, DOI 10.1090/S0002-9939-1972-0306174-8
E. W. Johnson and M. Detlefsen, Prime sequences and distributivity in local Noether lattices, Fund. Math. 86 (1974), 149–156. MR 414454, DOI 10.4064/fm-86-2-149-156