# Isotone maps on lattices

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## Abstract

Let \({\mathcal{L} = (Li | i \in I)}\) be a family of lattices in a nontrivial lattice variety **V**, and let \({\varphi_{i} : L_{i} \rightarrow M}\), for \({i \in I}\), be isotone maps (not assumed to be lattice homomorphisms) to a common lattice *M* (not assumed to lie in **V**). We show that the maps \({\varphi_{i}}\) can be extended to an isotone map \({\varphi : L \rightarrow M}\), where \({L = {\rm Free}_{V} \mathcal{L}}\) is the free product of the *L* _{ i } in **V**. This was known for **V = L**, the variety of all lattices. The above free product *L* can be viewed as the free lattice in **V** on the partial lattice *P* formed by the disjoint union of the *L* _{ i }. The analog of the above result does not, however, hold for the free lattice *L* on an arbitrary partial lattice *P*. We show that the only codomain lattices *M* for which that more general statement holds are the complete lattices. On the other hand, we prove the analog of our main result for a class of partial lattices *P* that are not-quite-disjoint unions of lattices. We also obtain some results similar to our main one, but with the relationship lattices : orders replaced either by semilattices : orders or by lattices : semilattices. Some open questions are noted.

## 2010 Mathematics Subject Classification

Primary: 06B25 Secondary: 06B20 06B23## Keywords and phrases

free product of lattices varieties prevarieties and quasivarieties of lattices isotone map free lattice on a partial lattice semilattice## Preview

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