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Freely adjoining a complement to a lattice. (English) Zbl 1164.06308
Summary: For a bounded lattice \(K\) and an element \(a\) of \(K - \{0,1\}\), we directly describe the structure of the lattice freely generated by \(K\) and an element \(u\) subject to the requirement that \(u\) be a complement of \(a\). Earlier descriptions of this lattice used multi-step procedures.
As an application, we give a short and direct proof of the classical result of R. P. Dilworth (1945): Every lattice can be embedded into a uniquely complemented lattice. We prove it in the stronger form due to C. C. Chen and G. Grätzer [J. Algebra 11, 56–63 (1969; Zbl 0185.03701)]: Every at most uniquely complemented bounded lattice has a \(\{0,1\}\)-embedding into a uniquely complemented lattice.

06B25 Free lattices, projective lattices, word problems
06C15 Complemented lattices, orthocomplemented lattices and posets
Full Text: EuDML
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