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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References:

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