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Bounded lattice expansions. (English) Zbl 0988.06003

The notion of a canonical extension of a lattice (with additional operations, respectively) is introduced. A canonical extension of a lattice \(L\) is a pair \((e,C)\), where \(C\) is a complete lattice, \(e:L\to C\) is a lattice embedding such that:
i) every element \(c\in C\) can be expressed both as a join of meets and a meet of joins of elements from \(e(L)\) (the image of \(L)\),
ii) whenever \(A\subset C\) is a set of elements which are joins of elements from \(e(L)\) and \(B\) is a set of elements which are meets of elements from \(e(L)\), then \(\bigwedge A\leq \bigvee B\) iff \(\bigwedge A^*\leq \bigvee B^*\) for some finite subsets \(A^*\subseteq A\) and \(B^*\subseteq B\).
Every lattice has a uniquely determined canonical extension. Both a concrete description and an abstract characterization of this extension is given. Various results involving the preservation of identities under canonical extension are obtained.

MSC:

06B05 Structure theory of lattices
06B23 Complete lattices, completions
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