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## Notes on sectionally complemented lattices. IV How far does the Atom Lemma go?

Acta Mathematica Hungarica
Authors:
G. Grätzer
and
M. Roddy

## Abstract

There are two results in the literature that prove that the ideal lattice of a finite, sectionally complemented, chopped lattice is again sectionally complemented. The first is in the 1962 paper of G. Gr�tzer and E. T. Schmidt, where the ideal lattice is viewed as a closure space to prove that it is sectionally complemented; we call the sectional complement constructed then the 1960 sectional complement. The second is the Atom Lemma from a 1999 paper of the same authors that states that if a finite, sectionally complemented, chopped lattice is made up of two lattices overlapping in an atom and a zero, then the ideal lattice is sectionally complemented. In this paper, we show that the method of proving the Atom Lemma also applies to the 1962 result. In fact, we get a stronger statement, in that we get many sectional complements and they are rather close to the componentwise sectional complement.

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## Notes on sectionally complemented lattices. III

Acta Mathematica Hungarica
Authors:
G. Grätzer
,
H. Lakser
, and
M. Roddy

Summary In a recent survey article, G. Grätzer and E. T. Schmidt raise the problem when is the ideal lattice of a sectionally complemented chopped lattice sectionally complemented. The only general result is a 1999 lemma of theirs, stating that if the finite chopped lattice is the union of two ideals that intersect in a two-element ideal U, then the ideal lattice of M is sectionally complemented. In this paper, we present examples showing that in many ways their result is optimal. A typical result is the following: For any finite sectionally complemented lattice U with more than two elements, there exists a finite sectionally complemented chopped lattice M that is (i) the union of two ideals intersecting in the ideal U; (ii) the ideal lattice of M is not sectionally complemented.

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