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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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