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Freely adjoining a relative complement to a lattice. (English) Zbl 1083.06011
Summary: Let $$K$$ be a lattice, and let $$a<b<c$$ be elements of $$K$$. We adjoin freely a relative complement $$u$$ of $$b$$ in $$[a,c]$$ to $$K$$ to form the lattice $$L$$. For two polynomials $$A$$ and $$B$$ over $$K \cup \{u\}$$, we find a very simple set of conditions under which $$A$$ and $$B$$ represent the same element in $$L$$, so that in $$L$$ all pairs of relative complements in $$[a,c]$$ can be described. Our major result easily follows: Let $$[a,c]$$ be an interval of a lattice $$K$$; let us assume that every element in $$[a,c]$$ has at most one relative complement. Then $$K$$ has an extension $$L$$ such that $$[a,c]$$ in $$L$$, as a lattice, is uniquely complemented.
As an immediate consequence, we get the classical result of R. P. Dilworth: Every lattice can be embedded into a uniquely complemented lattice. We also get the stronger form due to C. C. Chen and G. Grätzer: Every at most uniquely complemented bounded lattice has a $$\{0,1\}$$-embedding into a uniquely complemented lattice. Some stronger forms of these results are also presented.
A polynomial $$A$$ over $$K \cup \{u\}$$ naturally represents an element $$\langle A \rangle$$ of $$L$$. Let us call a polynomial A minimal, if it is of minimal length representing $$x$$. We characterize minimal polynomials.

##### MSC:
 06C15 Complemented lattices, orthocomplemented lattices and posets
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