Showing papers by "George Grätzer published in 2007"

Notes on planar semimodular lattices. I. Construction

Abstract: We construct all planar semimodular lattices in three simple steps from the direct product of two chains.

Subdirectly irreducible modular lattices of width at most 4

Abstract: In 1970, R. Freese proved that the variety M4 generated by modular lattices of width at most 4 has a finite basis. As an application, he obtained a complete description of all subdirectly irreducible members of this variety. We obtain an intuitive description of how congruences generated by a prime interval spread in a modular lattice of width at most 4, and apply the result to reprove Freese’s description of subdirectly irreducible lattices of width at most 4.

Notes on sectionally complemented lattices. IV. How far does the atom lemma go

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. Gratzer 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.