Notes on sectionally complemented lattices. IV How far does the Atom Lemma go? |
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Authors: | G Grätzer M Roddy |
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Institution: | (1) Department of Mathematics, University of Manitoba, Winnipeg, MB, R3T 2N2, Canada;(2) Department of Mathematics and Computer Science, Brandon University, Brandon, MB, R7A 6A9, Canada |
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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. Grätzer 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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Keywords: | sectionally complemented lattice chopped lattice ideal lattice congruence |
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