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Interpretations into \(p\)-algebras. (English) Zbl 0737.06010

A variety \(V\) of algebras is interpretable into a variety \(W\) (of algebras) iff the operations of \(V\) can be simulated by terms of \(W\). The paper under review proves that there are exactly two ways of interpreting the variety \(D_{01}\) of universally bounded distributive lattices into the variety \(B_ \omega\) of distributive lattices with pseudocomplementation (alias \(p\)-algebras): Either simulate join and meet in \(D_{01}\) by join and meet, respectively, in \(B_ \omega\) — the “trivial” interpretation — or use \(B_ \omega\)-join for \(D_{01}\)- meet and \(B_ \omega\)-meet for \(D_{01}\)-join. As a corollary one obtains that the trivial interpretation from \(B_ \omega\) into itself is the only possible one.
Reviewer: J.Schmid (Bern)

MSC:

06D15 Pseudocomplemented lattices
08B05 Equational logic, Mal’tsev conditions
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