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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.

##### MSC:
 06B25 Free lattices, projective lattices, word problems 06C15 Complemented lattices, orthocomplemented lattices and posets
##### Keywords:
relative complement; free lattice; uniquely complemented
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##### References:
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